Solving Integrals by Trigonometric substitution/Examples

Worked Example

Question

• $\int \frac{1}{x^2+1}dx$

If we look at the relationships between the things that are being substituted we can see that tan is the only one that has a 1 + ${\tan }^2\theta$. This means that it is best choice for use as a substition. The purpose of the substitution is to create denominator that is a single trigonometric identity. If we set $x^2$ equal to tan squared we get such an expression.

• $\int \frac{1}{{\tan }^{2}u+1}{\sec }^{2}udu$

Using the relationship.

• $=\int \frac{1}{{\sec }^{2}u}{\sec }^{2}udu$
• $=\int du$
• $=u+c$

Using the initial substitution

• $=\arctan x+c$

$\gamma =frac1\alpha +\beta$