Integrals: Problems with Solutions

By Hernando Guzman Jaimes (prof. at University of Zulia - Maracaibo, Venezuela)

Integral Formulas

$\int kdx=kx+C$   $k \in R$
$\int x^n dx=\frac{1}{n+1}x^{n+1}+C$   $n \ne -1$ $n \in Z$
$\int \frac{1}{x} dx=ln(x)+C$
$\int e^x dx=e^x+C$
$\int a^x dx=\frac{a^x}{ln(a)}+C$   $a \in R, a > 0$
$\int \sin(x) dx=-\cos(x)+C$
$\int \cos(x) dx=\sin(x)+C$
$\int \sec^2(x) dx=\tan(x)+C$
$\int \csc^2(x) dx=-\cot(x)+C$
$\int \sec(x)\tan(x) dx=\sec(x)+C$
$\int \csc(x)\cot(x) dx=-\csc(x)+C$
$\int \tan(x) dx=\ln(\sec(x))+C$
$\int \cot(x) dx=\ln(\sin(x))+C$
$\int \sec(x) dx=\ln(\sec(x) + \tan(x))+C$
$\int \csc(x) dx=\ln(\csc(x) - \cot(x))+C$
$\int \frac{dx}{\sqrt{a^2-x^2}} dx=\text{arcsin}\frac{x}{a}+C$   $a\in R$
$\int \frac{dx}{a^2+x^2} dx=\frac{1}{2}\text{arctan}\frac{x}{a}+C$   $a\in R$
$\int \frac{dx}{a^2-x^2} dx=\frac{1}{2a}\ln \left|\frac{x+a}{x-a}\right|+C$   $a\in R$

Integration Properties

$\int kf(x)dx = k\int f(x)dx$
$\int (f(x)\pm g(x))dx = \int f(x)dx \pm \int g(x)dx$

Problem 1
$g(x)=2x-3x^{2}$ is the derivative of $f(x)=x^{2}-x^{3}$.

$\int g(x)dx=?$
Problem 2
The derivative of the function $f(x)= \sqrt{x}+\frac{2\sqrt{x^{3}}}{3}$ is the function $g(x)=\sqrt{x}+\frac{1}{2\sqrt{x}}$.
Consider the following propositions:
(i) $\int g(x)dx=f(x)+C$

(ii) $\int f(x)dx=g(x)+C$

(iii) $\int \left( g(x)\right) ^{2}dx=\left(f(x)\right)^{2}+C$
Problem 3
$f(x)=\int \left( x^{2}+2\right) dx=\frac{1}{3}x^{3}+2x+C$ and $g(x)=\int \left( 2x^{2}-5\right) dx=\frac{2}{3}x^{3}-5x+C$
Evaluate the integral $\int \left( 3x^{2}-3\right) dx$ ?
Problem 4
If $f(x)=\int \left( 2x+3\right) dx=x^{2}+3x+C$ and $g(x)=\int \left( x^{2}+3\right) dx=\frac{1}{3}x^{3}+3x+C$

Evaluate $\int \left( 2x+3\right) \left( x^{2}+3\right) dx$ ?
Problem 5
How can we simplify the following integral?
$\int \frac{x^{2}+2x-3}{x^{4}}dx$ ?

Write the correct step by step solution.
Problem 6
Evaluate the integral
$\int (x+1)(x-2)dx=$
Problem 7
Evaluate the integral
$\int (2t^{2}-1)^{2}dt$
Problem 8
Evaluate the following integral $\int \sec y(\tan y-\sec y)dy$
Problem 9
Evaluate the integral
$\int \frac{\sin x}{1-\sin^{2}x}dx$
Problem 10
What is the best approximation by Riemann sums of the integral $f(x)=2x+5$ using $4$ rectangles in the interval $\left[0,2\right]$. Graph it.

Problem 11
What is the best approximation by Riemann sums of the integral $g(x)=2x^{2}-x-1$ using $6$ rectangles in the interval $\left[ 2,5\right]$.
Graph it.
Problem 12
What is the approximate area of the pink region, approximating the upper and lower sums.
Use $\Delta x=1$


Problem 13
What is the approximate area of the pink region, approximating the upper and lower sums.
Use $\Delta x=1$


Problem 14
Use upper and lower sums to approximate the area of the region under the curve $y=\sqrt{x}$.


Problem 15
Use upper and lower sums to approximate the area of the region under the curve $y=\frac{1}{x}$.


Problem 16
What is the result of the integral $\int \left( 2x+1\right) \left( x^{2}+x\right) dx$
Problem 17
Solve the integral using substitution.
$\int x^{2}\sqrt{x^{3}-5}dx$
Problem 18
Solve the integral using substitution.
$\int \frac{-4x}{\left( 1-2x^{2}\right) ^{2}}dx$
Problem 19
Solve the trigonometric integral $\int \cos ^{2}x\sin xdx$
Problem 20
Compute the definite integral $\underset{0}{\overset{1}{\int }} x\left( x^{2}+1\right) ^{3}dx$
Problem 21
Solve the definite integral $\overset{5}{\underset{1}{\int }} \frac{x}{\sqrt{2x-1}}dx$
Problem 22
What is the solution of this trigonometric integral?
$\int \tan ^{4}x\sec ^{2}xdx$
Problem 23
Solve this trigonometric integral
$\int \frac{\csc ^{2}x}{\cot ^{3}x}dx$
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