Integration by Trigonometric Substitution: Problems with Solutions

By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)
Problem 1
Which trigonometric substitution can we use to solve this integral?
$\int \frac{1}{\sqrt{\left(16-x^{2}\right) }}dx$
Problem 2
Solve the integral using subsitution.
$\int \frac{1}{x^{2}-25}dx=$
Problem 3
Solve the integral using substitution. Consider the image below.
$\int x\sqrt{1+x^{2}}dx = $


Problem 4
Solve this integral using substitution.
$\int \frac{9x}{\sqrt{1+x^{2}}}dx =$
Problem 5
Solve this integral by trigonometric substitution.
$\int \frac{1}{\sqrt{4x-x^{2}}}dx=$
Problem 6
$\int \frac{x^{2}}{\sqrt{16-x^{2}}}dx=$
Problem 7
Evaluate the integral by completing the square and using trigonometric substitution
$\int \frac{x}{\sqrt{x^{2}+6x+12}}dx=$
Problem 8
Evaluate the integral by completing the square and using trigonometric substitution
$\int \frac{x^{2}}{\sqrt{2x-x^{2}}}dx=$
Problem 9
To evaluate the integral $\int \frac{x-9}{x^{2}-6x}dx$ we do:
(i) factor the denominator
(ii) apply $(x^{2}-a^{2})=\left(x+a\right) \left( x-a\right)$
(iii) use partial fractions
Problem 10
Evaluate the integral $\int \frac{1}{x^{2}-9}dx$, using partial fractions.

Problem 11
Evaluate the integral $\int \frac{5}{x^{2}+3x-4}dx$, using partial fractions.
Problem 12
Evaluate the integral, using partial fractions.
$\int \frac{x+2}{x^{2}+11x+18}dx=$
Problem 13
How do we solve the integral?
$\int \frac{5x^{2}-12x-12}{x^{3}-4x} dx$

(i) factor the denominator and numerator
(ii) simplify the fraction
(iii) apply partial fractions
Problem 14
How do we solve the integral?
$\int \frac{x+2}{x^{2}-4x}dx$

(i) factor the denominator
(ii) simplify
(iii) apply partial fractions
Problem 15
How do we solve the integral? $\int \frac{x^{2}+2x+1}{x(x-1)(x+1)} dx$

(i) factor the numerator
(ii) simplify
(iii) apply partial fractions
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