Exercises

1. \( f(x) = 3x^7 \)
2. \( f(x) = \dfrac{5}{4}x^3 \)
3. \( f(x) = \dfrac{1}{2x^3} \)
4. \( f(x) = \dfrac{2}{3x} \)
5. \( f(x) = -\dfrac{4}{x^5} \)

1. \( f(x) = x^3 \)
2. \( f(x) = -\dfrac{10}{3}x^4 \)
3. \( f(x) = \dfrac{1}{2x^2} \)
4. \( f(x) = \dfrac{1}{2}e^x \)
5. \( f(x) = -\dfrac{3}{x} \)

1. \( f(x) = x^3 + \sqrt{x} \)
2. \( f(x) = 3e^{2x} \)
3. \( f(x) = \dfrac{1}{\sqrt{x}} \)
4. \( f(x) = 2\cos(x) \)
5. \( f(x) = \sin(3x) + x^2 \)

1. Calculate the rectangular approximations \( R_1 \), \( R_2 \), and \( R_4 \) for \( f(x) = 2x - 1 \) over the interval \( [1, 5] \).
2. Calculate the exact value of the area under \( f(x) = 2x - 1 \) over the interval \( [1, 5] \) by sketching the function. Compare this answer to your approximations from a).

1. Calculate the rectangular approximations \( L_1 \), \( L_3 \), and \( L_6 \) for \( f(x) = 3x + 4 \) over the interval \( [-1, 2] \).
2. Calculate the exact value of the area under \( f(x) = 3x + 4 \) over the interval \( [-1, 2] \) by sketching the function. Compare this answer to your approximations from a).

1. \( R_4 \) for \( f(x) = x^2 \) over \( [0, 1] \)
2. \( M_4 \) for \( f(x) = 1 - x^2 \) over \( [0, 1] \)

1. \( M_4 \) for \( f(x) = e^x \) over \( [-1, 1] \)
2. \( L_6 \) for \( f(x) = e^{-x} \) over \( [0, 1] \)

1. Suppose that a cyclist is riding on a straight road and the velocity of the bike at various times is recorded in the following table:

   \( t \) (in hours)	\( 0 \)	\( 0.4 \)	\( 0.8 \)	\( 1.2 \)	\( 1.6 \)	\( 2.0 \)
   \( v(t) \) (in km/h)	\( 20 \)	\( 18 \)	\( 15 \)	\( 19 \)	\( 18 \)	\( 14 \)

   Calculate \( L_5 \) for \( v(t) \) over \( [0, 2] \) and explain its physical meaning.
2. A group takes a road trip from Kelowna to Radium Hot Springs, \( 451 \) km away according to their map. To pass the time, the passengers note the speed of the car every few minutes using the odometer, and find that the velocity of the car, in km/h at time \( t \), in hours, can be approximated by the following continuous curve:
   Approximate \( R_{10} \) and \( L_{10} \) of \( v(t) \) over \( [0, 5] \) and interpret their meaning physically.

1. Calculate \( R_4 \) for \( f(x) \) over \( [0, 1]. \)
2. Write \( R_n \) as a function of \( n \).
3. Find \( \displaystyle \lim_{n \rightarrow \infty} R_n \).
4. Can you explain your answer from part c) using the formula for the area of a particular triangle?

1. Calculate \( L_5 \) and \( R_5 \) for \( f(x) \) over \( [0, 1]. \)
2. Write \( L_n \) and \( R_n \) as a function of \( n \).
3. Find \( \displaystyle \lim_{n \rightarrow \infty} L_n \) and \( \displaystyle \lim_{n \rightarrow \infty} R_n \).

1. \( 1^2 + 2^2 + 3^2 + 4^2 + 5^2 \)
2. \( \sqrt{4} + \sqrt{5} + \sqrt{6} + \cdots + \sqrt{10} \)
3. \( 1^3 + 2^3 + 3^3 + \cdots + n^3 \)
4. \( 7 + 9 + 11 + \cdots + 17 \)
5. \( 2 + 4 + 8 + 16 + \cdots + 128 \)

1. \( \displaystyle \sum_{k=1}^{5} (7k - 2) \)
2. \( \displaystyle \sum_{k=3}^{7} 1^k \)
3. \( \displaystyle \sum_{k=5}^{6} k^3 \)
4. \( \displaystyle \sum_{k=1}^{10} 3 \)
5. \( \displaystyle \sum_{k=2}^{6} (k^2 + k) \)

1. Calculate the Riemann sums \( R_1 \), \( R_2 \), and \( R_4 \) for \( f(x) = 2x - 3 \) over the interval \( [0, 4] \).
2. Find the exact value of the net area bounded by \( f(x) \) and the \( x \)-axis over the given interval by sketching the function. Compare this answer to your approximations from a).

1. Calculate the Riemann sums \( M_1 \), \( M_3 \), and \( M_6 \) for \( f(x) = 1 - 3x \) over the interval \( [-1, 2] \).
2. Find the exact value of the net area bounded by \( f(x) \) and the \( x \)-axis over the given interval by sketching the function. Compare this answer to your approximations from a).

1. Sketch the function \( f(x) \) over the interval \( [-1, 2 ] \) by first writing this polynomial in factored form.
2. How many regions are bounded by the function \( f(x) \) and the \( x \)-axis over \( [-1, 2] \)? Are they positive areas or negative areas?
3. Compute the Riemann sum \( M_3 \) for \( f(x) \) over the interval \( [-1, 0] \). What is this Riemann sum approximating?
4. Compute the Riemann sum \( M_4 \) for \( f(x) \) over the interval \( [0, 2] \). What is this Riemann sum approximating?
5. Compute the Riemann sum \( M_6 \) for \( f(x) \) over the interval \( [-1, 2] \). What is this Riemann sum approximating?

• a lower bound for the integral
• an upper bound for the integral
• the best approximation for the integral

1. Evaluate the following integrals by finding the limit of Riemann sums.
   1. \( \displaystyle \int_{0}^{2} (x^{2} - 2x)~dx \)
   2. \( \displaystyle \int_{0}^{3} (x^{2} - 2x)~dx \)
2. What do the values of these two integrals tell you about the "signed" areas bounded by the curve \( f(x) = x^{2}-2x \) and the \( x \)-axis over \( [0,3] \)?

1. \( \displaystyle \int_{1}^{3} (x-3)~dx \)
2. \( \displaystyle \int_{3}^{6} (x-3)~dx \)
3. \( \displaystyle \int_{0}^{6} (x-3)~dx \)

1. \( \displaystyle \int_{0}^{2} \sqrt{4-x^{2}}~dx \)
2. \( \displaystyle \int_{-3}^{3} \sqrt{9-x^{2}}~dx \)
3. \( \displaystyle \int_{-1}^{0} -\sqrt{1-x^{2}}~dx \)

1. The area below the curve \( y=\cos(x) \) and above the \( x \)-axis over the interval \( \left[ -\frac{\pi}{2} , \frac{\pi}{2} \right] \).
2. The area below the curve \( y= 9-x^{2} \) and above the \( x \)-axis.
3. The area above the curve \( y=x^{2}-4 \) and below the \( x \)-axis.

1. \( \displaystyle \int_{-1}^{3} (2-\left \lvert x \right \rvert)~dx \)
2. \( \displaystyle \int_{0}^{5} (\left \lvert x-3 \right \rvert-1)~dx \)
3. \( \displaystyle \int_{-2}^{2}\left(1+ \sqrt{4-x^{2}}\right)~dx \)

1. \( \displaystyle \int_{-1}^{1} x^{3} ~dx \)
2. \( \displaystyle \int_{-1}^{1} |x|^{3} ~dx \)
3. \( \displaystyle \int_{-2}^{-1} -(x+2)^{3} ~dx \)

1. \( \displaystyle \int_{0}^{7}f(x)~dx \)
2. \( \displaystyle \int_{0}^{4}f(x)~dx \)
3. \( \displaystyle \int_{4}^{4}f(x)~dx \)
4. \( \displaystyle \int_{4}^{0}f(x)~dx \)
5. \( \displaystyle \int_{4}^{7}f(x)~dx \)
6. \( \displaystyle \int_{0}^{3}f(x)~dx \)

1. \( \displaystyle \int_{0}^{1} (x+1)~ dx + \int_{1}^{3} (x+1) ~dx \)
2. \( \displaystyle \int_{1}^{3}\cos(x)~dx + 3 \int_{1}^{2} \sin(x)~dx+ \int_{3}^{1} \cos(x)~dx \)
3. \( \displaystyle \int_{-1}^{1} 3x^{3}~dx + \int_{-1}^{1} x~dx + \int_{-1}^{1} \frac{1}{2}(x+3x^{3})~dx \)

1. \( \displaystyle \int_{0}^{1} f(x)~dx + \int_{-1}^{0} f(x)~dx + \int_{1}^{-1} \frac{1}{3}f(x)~dx \)
2. \( \displaystyle \int_{2}^{4} f(x)~dx + \int_{0}^{2} 4 f(x)~dx - \int_{3}^{4} f(x)~dx + \int_{3}^{3} \pi f(x)~dx \)
3. \( - \displaystyle \int_{3}^{4} f(x)~dx + \int_{7}^{4} 5 f(x)~dx + \int_{3}^{7} f(x)~dx \)

1. \( \displaystyle \int_{-1}^{1} 2 x^{4}~dx \)
2. \( \displaystyle \int_{-1}^{1} (4x^{2} +10 x^{4})~dx \)
3. \( \displaystyle \int_{-1}^{1} (5x^{4} - 3x^{2})~dx \)

1. \( \displaystyle \int_{1}^{0} 2e^{x} ~dx \)
2. \( \displaystyle \int_{0}^{2} -4 x^{2} ~dx \)
3. \( \displaystyle \int_{0}^{1} (e^{x} + \pi x^{2}) ~dx \)
4. \( \displaystyle \int_{1}^{2} x^{2} ~dx \)
5. \( \displaystyle \int_{0}^{1} (e^{x+1} + 2x^{2}) ~dx \)

1. Use this property to show that if \( m \leq f(x) \leq M \) for all \( x \) in the interval \( [a, b] \), then\[ m(b - a) \leq \int^{b}_{a} f(x) ~dx \leq M(b - a) \]
2. Suppose that \( 1 \leq f(x) \leq 2 \) for all \( x \in [-2, 2] \). Verify the conclusion from part a) for this specific case, using an area interpretation of the three quantities in the inequality.
3. Use part a) to justify the following inequality:\[ \frac{\pi}{2} \leq \int^{\frac{\pi}{2}}_{0} \left( \cos^3{(x)} + 1 \right)~dx \leq \pi \]

1. Consider the following graph of a continuous function \( f(t) \) on \( 0 \leq t \leq 10 \).
   Let \( F(x) = \displaystyle \int^{x}_{0} f(t)~dt \). Determine the following values of \( F(x) \):
   1. \( F(0) \)
   2. \( F(4) \)
   3. \( F(8) \)
   4. \( F(10) \)
2. Let \( g(t) \) be the function whose graph is given below. Let \( G(x) = \displaystyle \int^{x}_{0} g(t) ~ dt, 0 \leq x \leq 5 \). Where does \( G(x) \) attain its maximum and minimum values over \( [0, 5] \)? What are the maximum and minimum values?
   

1. \( \displaystyle \int^{2}_{0} 2x ~ dx \)
2. \( \displaystyle \int^{3}_{1} 4x^3 ~ dx \)
3. \( \displaystyle \int^{1}_{0} 5e^x ~ dx \)
4. \( \displaystyle \int^{4}_{0} \left( \sqrt{x} + 2x \right) ~ dx \)

1. \( \displaystyle \int^{1}_{-1} 3 ~ dx \)
2. \( \displaystyle \int^{2}_{1} -\tfrac{1}{2} x^2 ~ dx \)
3. \( \displaystyle \int^{5}_{-1} \left( \frac{2}{x^4} - x^4 \right) ~ dx \)
4. \( \displaystyle \int^{-1}_{-2} \dfrac{1}{x} ~ dx \)

1. \( g(x) = \displaystyle \int^{x}_{0} \dfrac{1}{1 + t^2} ~ dt \)
2. \( g(x) = \displaystyle \int^{x}_{-1} \left( e^{-t} - \cos(2t) \right) ~ dt \)
3. \( g(x) = \displaystyle \int^{x}_{5} \left[ \ln\left( \frac{2}{t} \right) + 1 \right] ~ dt \)

1. \( h(x) = \displaystyle \int^{2x}_{0} \dfrac{1}{1 + t^2} ~ dt \)
2. \( h(x) = \displaystyle \int^{2}_{x^2} \sin(t) ~ dt \)
3. \( h(x) = \displaystyle \int^{\cos(x)}_{1} \left( 3 - t^2 \right) ~ dt \)

1. \( g(x) = \displaystyle \int^{x}_{1} t^5 ~ dt \), \( x = 2 \)
2. \( g(x) = \displaystyle \int^{x}_{0} e^{-t} ~ dt \), \( x = 1 \)

1. \( \displaystyle \int^{b}_{a} 1 ~dx \) 
2. \( \displaystyle \int^{b}_{a} 3x^2 ~ dx \)
3. \( \displaystyle \int^{2b}_{a} \dfrac{1}{x^2} ~ dx \)

1. Explain why \( F \) must attain an absolute maximum and minimum value over \( [0, 8] \).
2. Locate and classify any local extremes of \( F \).
3. Estimate the location of any inflection points of \( F \).
4. Give a rough sketch of the function \( F(x) \) for \( 0 \leq x \leq 8 \) by estimating areas.

1. \( \displaystyle \int \sin(\pi x) ~ dx = -\frac{1}{\pi} \cos(\pi x) + c \)
2. \( \displaystyle \int \dfrac{1}{\sqrt{x + 2}} ~ dx = 2\sqrt{x + 2} + c \)
3. \( \displaystyle \int 2\left( e^{3x} - x \right) ~ dx = \frac{2}{3}e^{3x} - x^2 + c \)
4. \( \displaystyle \int xe^{x} ~ dx = (x - 1)e^x + c \)

1. \( \displaystyle \int \left( 3x + 1 \right)^3 ~ dx = \frac{1}{12} \left( 3x + 1 \right)^4 + c \)
2. \( \displaystyle \int x \sin(x) ~ dx = -\frac{x^2}{2} \cos(x) + c \)
3. \( \displaystyle \int xe^{2x} ~ dx = xe^{2x} - \frac{e^{2x}}{2} + c \)
4. \( \displaystyle \int x\cos(x) ~ dx = \cos(x) + x \sin(x) - 2 + c \)

1. \( \displaystyle \int x ~ dx \)
2. \( \displaystyle \int x^4 ~ dx \)
3. \( \displaystyle \int \frac{1}{x^2} ~ dx \)
4. \( \displaystyle \int \sqrt{x^3} ~ dx \)

1. \( \displaystyle \int \left( x^2 + x \right) ~ dx \)
2. \( \displaystyle \int \left( 1 - x^3 \right) ~ dx \)
3. \( \displaystyle \int \left( x^2 - 2x + 2 \right) ~ dx \)
4. \( \displaystyle \int \left( 5x^4 + 3x^2 + 2x \right) ~ dx \)

1. \( \displaystyle \int \left( \sin(x) + \cos(x) \right) ~ dx \)
2. \( \displaystyle \int \left( 2\sec^2(x) - \sin(x) \right) ~ dx \)
3. \( \displaystyle \int \left( \sec(x)\tan(x) - 1 \right) ~ dx \)
4. \( \displaystyle \int \csc(x) \left( \cot(x) + \csc(x) \right) ~ dx \)

1. \( \displaystyle \int \left( \sqrt{x} + 2e^x \right) ~ dx \)
2. \( \displaystyle \int \left( 3 \sin(x) - e^x + 1 \right) ~ dx \)
3. \( \displaystyle \int \left( 2^x + 3^x \right) ~ dx \)

1. \( \displaystyle \int \left( 2x - \sqrt[3]{x^2} \right) ~ dx \)
2. \( \displaystyle \int \left( e^t - 2\cos(t) + \frac{1}{t^2} \right) ~ dt \)
3. \( \displaystyle \int \left( x^5 + \frac{3}{x} - \frac{1}{3} \sec^2(x) \right) ~ dx \)

1. \( \displaystyle \int \dfrac{2e^{2x} + e^x x^3}{2e^x} ~ dx \)
2. \( \displaystyle \int \dfrac{\sin(t) + \cos^2(t)}{\cos^2(t)} ~ dt \)
3. \( \displaystyle \int \dfrac{\sqrt{u} - u^2 \sin{(u)} + 5}{u^2} ~ du \)

1. \( \displaystyle \int \left( \frac{1}{x} - \frac{1}{x^2} + \frac{1}{x^3} \right) ~ dx \)
2. \( \displaystyle \int 3^{u + 1} ~ du \)
3. \( \displaystyle \int \left( \dfrac{x \csc^2(x) + 2}{x} \right) ~ dx \)
4. \( \displaystyle \int \left( \dfrac{\cos(\theta) + \sin^{3}(\theta)}{\sin^2(\theta)} \right) ~ d\theta \)
5. \( \displaystyle \int \left( e^{-2t} + \cos(\pi t) - \sin(\pi t) \right) ~ dt \)
6. \( \displaystyle \int 2^{x - 1} ~ dx \)

1. For all \( k \in \mathbb{R} \) and all continuous functions \( f \), \( \displaystyle \int k f(x) ~ dx = k \int f(x) ~ dx \)
2. For all continuous functions \( f \) and \( g \), \( \displaystyle \int \left( f(x) + g(x) \right) ~ dx = \int f(x) ~ dx + \int g(x) ~ dx \)

1. \( \displaystyle \int^{2}_{0} f(x) ~ dx \)
2. \( \displaystyle \int^{3}_{-1} f(x) ~ dx \)
3. \( \displaystyle \int^{3}_{0} 2f(x) ~ dx \)
4. \( \displaystyle \int^{1}_{2} f(x) ~ dx \)
5. \( \displaystyle \int^{3}_{1} \left( f(x) - 4 \right) ~ dx \)

1. \( \displaystyle \int^{2}_{1} \left( 2x + 3 \right) ~ dx \)
2. \( \displaystyle \int^{3}_{0} \left( x - x^2 \right) ~ dx \)
3. \( \displaystyle \int^{1}_{-1} \left( x^5 + 1 \right) ~ dx \)
4. \( \displaystyle \int^{1}_{0} \left( 4x^3 - 12x^5 + 3x^2 \right) ~ dx \)

1. \( \displaystyle \int^{\frac{\pi}{2}}_{0} \big( \cos(x) + \sin(x) \big) ~ dx \)
2. \( \displaystyle \int^{\frac{\pi}{4}}_{0} \big( \sec^2(x) + 1 \big) ~ dx \)
3. \( \displaystyle \int^{\frac{\pi}{3}}_{0} \big( \sec(x) \tan(x) - \sin(x) \big) ~ dx \)

1. \( \displaystyle \int^{1}_{0} 4^x ~ dx \)
2. \( \displaystyle \int^{1}_{-1} \left( x - e^x \right) ~ dx \)
3. \( \displaystyle \int^{2}_{1} \left( 2^x + 3^x \right) ~ dx \)

1. \( f(x) = x^2 - \dfrac{1}{x^2} \), \( x \in [1, 3] \)
2. \( f(x) = 2x - e^{-2x} \), \( x \in \left[ -\frac{1}{2}, 0 \right] \)
3. \( f(t) = e^t + \cos(\pi t) \), \( t \in [0, 1] \)

1. \( \displaystyle \int^{\frac{\pi}{3}}_{0} \dfrac{\tan(x) + \cos^2(x)}{\cos(x)} ~ dx \)
2. \( \displaystyle \int^{1}_{-1} \dfrac{2e^{2t} - 1}{e^t} ~ dt \)
3. \( \displaystyle \int^{1}_{\frac{\pi}{4}} \dfrac{\cot(\theta)}{\sin(\theta)\cos(\theta)} ~ d\theta \)

1. \( \displaystyle \int^{\frac{\pi}{3}}_{\frac{\pi}{6}} \big( \sin(u) - 2\cos(u) \big) ~ du \)
2. \( \displaystyle \int^{-2}_{-1} \dfrac{3x^2 + x^4}{x^2} ~ dx \)
3. \( \displaystyle \int^{\frac{\pi}{2}}_{\frac{\pi}{4}} \csc(x)\cot(x) ~ dx \)
4. \( \displaystyle \int^{1}_{-1} \Big( e^{-t} - \sin{(\pi t)} \Big) ~ dt \)

1. \( \displaystyle \int \sin(2x + 1) ~ dx \), \( u = 2x + 1 \)
2. \( \displaystyle \int \sqrt{3 - x} ~ dx \), \( u = 3 - x \)
3. \( \displaystyle \int e^{x - 1} ~ dx \), \( u = x - 1 \)
4. \( \displaystyle \int \sec^2\left( 2\pi x + 2 \right) ~ dx \), \( u = 2\pi x + 2 \)

1. \( \displaystyle \int \cos(7x - 5) ~ dx \)
2. \( \displaystyle \int 5(x + 3)^4 ~ dx \)
3. \( \displaystyle \int \sqrt{1 - 5x} ~ dx \)
4. \( \displaystyle \int \dfrac{1}{\cot(6x)} ~ dx \)

1. \( \displaystyle \int x \cos\left( 3x^2 \right) dx \), \( u = 3x^2 \)
2. \( \displaystyle \int \sec(4\theta)\tan(4\theta) ~ d\theta \), \( u = 4\theta \)
3. \( \displaystyle \int \left( x^5 + 5x^3 + 1 \right) \left( x^4 + 3x^2 \right) dx \), \( u = x^5 + 5x^3 + 1 \)
4. \( \displaystyle \int \dfrac{\pi r^2}{\sqrt{1 - 2r^3}} ~ dr \), \( u = 1 - 2r^3 \)

1. \( \displaystyle \int 2e^x \cos{(e^x)} ~ dx \)
2. \( \displaystyle \int \big( 2 \tan(\theta) + 4 \big)^5 \sec^2(\theta) ~ d\theta \)
3. \( \displaystyle \int 4 \left( 3x^2 + 2 \right)\left( x^3 + 2x \right)^3 ~ dx \)

• Use the method of substitution to find an antiderivative of the integrand and then use the fundamental theorem to evaluate.
• Use substitution to rewrite the entire integral, including the limits of integration, in terms of a new variable \( u \), and then evaluate.

1. \( \displaystyle \int^{1}_{0} 3x \sqrt{x^2 + 1} ~ dx \), \( u = x^2 + 1 \)
2. \( \displaystyle \int^{\frac{\pi}{3}}_{\frac{\pi}{6}} \sin^3(x) \cos(x) ~ dx \), \( u = \sin(x) \)
3. \( \displaystyle \int^{1}_{-1} x^3 \left( x^4 - 2 \right)^3 ~ dx \), \( u = x^4 - 2 \)
4. \( \displaystyle \int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \cos(x) \left( 4 + \frac{1}{3} \sin(x) \right)^2 ~ dx \), \( u = 4 + \frac{1}{3} \sin(x) \)

1. \( \displaystyle \int^{1}_{-1} 2x \left( x^2 - 1 \right)^2 ~ dx \)
2. \( \displaystyle \int^{\frac{\pi}{4}}_{0} \big( \tan(\theta) \sec(\theta) \big)^2 ~ d\theta \)
3. \( \displaystyle \int^{e}_{1} \dfrac{\ln(x)}{x} ~ dx \)
4. \( \displaystyle \int^{1}_{0} 3x e^{x^2} ~ dx \)
5. \( \displaystyle \int^{\frac{\pi^2}{16}}_{\frac{\pi^2}{9}} \dfrac{\sin{(\sqrt{x})}}{\sqrt{x}} ~ dx \) (hint: let \( u = \sqrt{x} \))

1. Derive the identity\[ \cos(x)\cos(y) = \frac{1}{2} \big[ \cos(x + y) + \cos(x - y) \big] \] using the sum and difference formulas for cosine.
2. Let \( m \) and \( n \) be distinct non-negative integers. Verify the identity\[ \int^{2\pi}_{0} \cos(mx) \cos(nx) ~ dx = 0 \] Indicate where in your solution you are using the fact that \( m \) and \( n \) are distinct, that \( m \) and \( n \) are non-negative, and that \( m \) and \( n \) are integers.
3. Let \( m \) be any integer. Verify the following:\[ \int^{2\pi}_{0} \cos^2(mx) ~ dx = \begin{cases} 2\pi & \text{if } m = 0 \\ \pi & \text{if } m \neq 0 \end{cases} \]

In this exercise, you will prove the substitution rule for indefinite integrals using the following steps.

Let \( f \) be a continuous function with \( F' = f \) and let \( g \) be any continuous function.

1. Explain why \( \displaystyle \int F'(g(x)) g'(x) ~ dx = F(g(x)) + c \).
2. Explain why \( \displaystyle \int F'(u) ~ du = F(u) + c \) for any variable \( u \).
3. Let \( u = g(x) \). Prove the substitution rule\[ \int f(g(x))g'(x) ~ dx = \int f(u) ~ du \] using the results from part a) and part b).

1. Determine \( \displaystyle \int x \sin(3x) ~ dx \) using integration by parts with \( u = x \) and \( dv = \sin(3x) ~ dx \).
2. Determine \( \displaystyle \int 4x \ln(x) ~ dx \) using integration by parts with \( u = \ln(x) \) and \( dv = 4x ~ dx \).
3. Determine \( \displaystyle \int te^{-t} ~ dt \) using integration by parts with \( u = t \) and \( dv = e^{-t} ~ dt \).

1. \( \displaystyle \int (2x + 1) \cos(x) ~ dx \)
2. \( \displaystyle \int \cos(\theta) \ln( \sin(\theta) ) ~ d\theta \)
3. \( \displaystyle \int x \sec^2(x) ~ dx \)

1. \( \displaystyle \int t^2 e^{t} ~ dt \)
2. \( \displaystyle \int (x^3 + 1) \ln(x) ~ dx \)

1. \( \displaystyle \int^{4}_{1} \ln(\sqrt{x}) ~ dx \)
2. \( \displaystyle \int^{e}_{1} 3\left( \ln(x) \right)^2 ~ dx \)
3. \( \displaystyle \int^{1}_{0} \left( t^2 + t \right) e^t ~ dt \)

1. \( \displaystyle \int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} (x - 1)\sin(\pi x) ~ dx \)
2. \( \displaystyle \int^{2\pi}_{0} \theta^2 \cos(2 \theta) ~ d\theta \)

In this question, you will use integration to calculate the centroid (or centre of mass) of regions in the plane. If you think of a two-dimensional region as a thin, uniformly dense solid, then the centroid of the region is located at the balancing point of this object.

Let \( R \) be the region bounded by a continuous function \( f(x) \geq 0 \) over an interval \( [a, b] \), and let \( A \) denote the area of the region \( R \). The \( x \) and \( y \) coordinates of the centroid of \( R \) are calculated as follows:

1. Using the given formulas, determine the location of the centroid of the region bounded by the function \( f(x) = 1 \) over \( [0, 2] \). Does this answer agree with what one would expect from looking at the region in question?
2. Using the given formulas, determine the location of the centroid of the region bounded by the function \( f(x) = \sqrt{1 - x^2} \) over \( [-1, 1] \). Does this answer agree with what one would expect from looking at the region in question?
3. Determine the location of the centroid of the region bounded by the function \( f(x) = \ln(x) \) over \( [1, e] \).

1. Use integration by parts to show that\[ \lim_{t \to \infty} \int^{t}_{0} x^n e^{-x} ~ dx = n \lim_{t \to \infty} \int^{t}_{0} x^{n - 1} e^{-x} ~ dx \] for all positive integers \( n \).
2. Using the result from part a), determine the value of \( \displaystyle \lim_{t \to \infty} \int^{t}_{0} x^5 e^{-x} ~ dx \).