Exercises 1

This page allows you to practice some exercises on Integration. If you notice that you have difficulties, we advise you to go over the corresponding sections in the book.

These questions correspond to the following sections: - Section 10.1 - Indefinite Integrals - Section 10.2 - Area and Definite Integrals

Question 1

Find the following integrals.

(a) $\int x^{13} \text{ d}x$

(b) $\int x\sqrt{x} \text{ d}x$

(c) $\int \sqrt{x \sqrt{x \sqrt{x}}} \text{ d}x$

(d) $\int e^{x/4} \text{ d}x$

(e) $\int 3e^{-2x} \text{ d}x$

(f) $\int 2^{x} \text{ d}x$

(g) $\int (x^{3} + 2x - 3) \text{ d}x$

(h) $\int (x-1)^{2} \text{ d}x$

Show answer

(a) $\frac{1}{14}x^{14} + C$.

(b) $\frac{2}{5}x^2\sqrt{x} + C$.

(c) $\frac{8}{15}x^{5/18} + C$, by exploiting the fact that $\sqrt{x\sqrt{x \sqrt{x}}} = x^{7/8}$.

(d) $\frac{1}{4}e^{\frac{1}{4}x} + C$.

(e) $-\frac{3}{2}e^{-2x} + C$.

(f) $(1/\ln 2)2^{x} + C$.

(g) $\frac{1}{4}x^{4} + x^{2} - 3x + C$.

(h) $\frac{1}{3}(x-1)^{3} + C$.

Question 2

Find the following integrals.

(a) $\int (x-1)(x+2) \text{ d}x$

(b) $\int \frac{x^{3} - 3x + 4}{x} \text{ d}x$

(c) $\int \frac{(y - 2)^2}{\sqrt{y}} \text{ d}y$

(d) $\int \frac{x^{3}}{x + 1} \text{ d}x$

(e) $\int x(1+x^2)^{15} \text{ d}x$

Hints: In part (c), first expand the numerator, and then divide each term by the denominator. In part (d), use polynomial division. In part (e), what is the derivative of $(1+x^2)^{16}$?

Show answer

(a) $\frac{1}{3}x^3 + \frac{1}{2}x^2 - 2x + C$.

(b) $\frac{1}{3}x^3 - 3x + 4 \ln |x| + C$.

(c) $\frac{2}{5}y^2\sqrt{y} - \frac{8}{3} y\sqrt{y} + 8\sqrt{y} + C$.

(d) $\frac{1}{3}x^3 - \frac{1}{2} x^2 + x - \ln |x+1| + C$, as $x^3/(x+1) = x^2 - x + 1 - 1/(x+1)$.

(e) $\frac{1}{32} (1 + x^2)^{16} + C$.

Question 3

Provided $a \neq 0$ and $p \neq -1$, show that $\int (ax +b)^{p} \text{ d}x = \frac{1}{a(p+1)} (ax + b)^{p+1} + C$.

Show answer

The result follows directly by differentiating the right-hand side. You obtain the required integrand.

Another way to show the result is to actually evaluate the integral, which can be done using integration by substitution.

Question 4

Find $F(x)$ if:

(a) $F'(x) = \frac{1}{2}e^{x} - 2x$ and $F(0) = \frac{1}{2}$.

(b) $F'(x) = x(1 - x^{2})$ and $F(1) = \frac{5}{12}$.

Show answer

(a) $F(x) = \int \left( \frac{1}{2} e^{x} - 2x \right) \text {d}x = \frac{1}{2} e^{x} - x^2 + C$. Then $F(0) = 1/2$ implies $C=0$.

(b) $F(x) = \int (x - x^{3}) \text{ d}x = \frac{1}{2}x^2 - \frac{1}{4} x^4 + C$. Then $F(1) = 5/12$ implies $C = 1/6$.

Question 5

Suppose that $f''(x) = x^{-2} + x^{3} + 2$ for $x > 0$, and $f(1) = 0, f'(1) = 1/4$. Find $f(x)$.

Show answer

$f(x) = -\ln x + \frac{1}{20} x^{5} + x^{2} - x - \frac{1}{20}$.

Question 6

Compute the area bounded by the graph of each of the following functions over the indicated interval.

(a) $f(x) = 3x^2,$ in $[0,2]$

(b) $f(x) = 1/x^2$, in $[1, 10]$

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(a) $\int_{0}^{2} 3x^2 \text{ d}x = \Big|_{0}^{2} x^{3} = 8 - 0 = 8.$

(b) $\int_{1}^{10} 1/x^{2} = \Big|_{1}^{10} (-1/x) = -1/10 - (- 1) = 9/10$.

Question 7

Compute the area bounded by the graph of $f(x) = 1/x^3$, the $x$-axis, and the two lines $x= -2$ and $x= -1$. Hint: $f(x) < 0$ in $[-2, -1]$.

Show answer

$A = -\int_{-2}^{-1} x^{-3} \text {d}x = - \Big|_{-2}^{-1} \left(-\frac{1}{2} \right) x^{-2} = -\left[-\frac{1}{2} - \left(-\frac{1}{8} \right) \right] = \frac{3}{8}$.

Question 8

Evaluate the following integrals:

(a) $\int_{1}^{2} (2x + x^2) \text{ d}x$

(b) $\int_{-2}^{3} \left( \frac{1}{2}x^{2} - \frac{1}{3}x^{3} \right) \text{ d}x$

(c) $\int_{2}^{3} \left( \frac{1}{t-1} + t \right) \text{ d}t$

Show answer

(a) $\Big|_{1}^{2} x^2 + \frac{1}{3}x^{3} = \left(4 + \frac{8}{3} \right) - \left(1 + \frac{1}{3} \right) = 16/3.$

(b) $\Big|_{-2}^{3} \frac{1}{6}x^{3} - \frac{1}{12} x^{4} = -\frac{27}{12} - \left(-\frac{32}{12} \right) = 5/12.$

(c) $\Big|_{2}^{3} \ln(t-1) + \frac{1}{2}t^3 = \left(\ln 2 + \frac{9}{2} \right) - \left(\ln1 + \frac{4}{2} \right) = \ln 2 + 5/2, \text{ as } \ln 1 = 0.$

Question 9

Let $f(x) = x(x-1)(x-2)$.

(a) Calculate $f'(x)$. Where is $f(x)$ increasing?

(b) Calculate $\int_{0}^{1} f(x) \text{ d}x$.

Show answer

(a) Since $f(x) = x^3 - 3x^2 + 2x$, we obtain $f'(x) = 3x^2 - 6x + 2$. Setting this derivative equal to zero, we obtain $x_{0} = 1 - \frac{1}{3}\sqrt{x}$ and $x_{1} = 1 + \frac{1}{3}\sqrt{3}$. So, $f(x)$ increases in $(-\infty, x_{0})$and in $(x_{1}, \infty)$.

(b) $\int_{0}^{1} x^3 - 3x^2 + 2x = \Big| \frac{1}{4}x^4 - x^3 + x^2 = \left( \frac{1}{4} - 1 + 1 \right) - 0 = \frac{1}{4}$. You could also graph this to get some insights in what area you have exactly calculated.