Question

Evaluate the indefinite integral, using trigonometric substitution and a triangle to express the answer in terms of x.

Answer 1

Given the indefinite integral:

$\int \frac{1}{(25+4x^2)^{\frac{3}{2}}}dx\text{ }$

Applying integration by trigonometric substitution:

$\int \frac{1}{50\sec (u)}du$$=\frac{1}{50}\int \frac{1}{\sec (u)}du$$=\frac{1}{50}\int \cos (u)du$$=\frac{1}{50}\sin (u)$

Now we substitute into the equation u = arctan(2/5x)

$=\frac{1}{50}\sin (\arctan (\frac{2}{5}x))$

Finally simplifying:

$=\frac{x}{25(4x^2+25)^{\frac{1}{2}}}+\text{ C}$