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)^{(3)/(2)}}dx\text{ }`

Applying integration by trigonometric substitution:

`\int (1)/(50\sec (u))du`
`=(1)/(50)\int (1)/(\sec (u))du`
`=(1)/(50)\int \cos (u)du`
`=(1)/(50)\sin (u)`

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

`=(1)/(50)\sin (\arctan ((2)/(5)x))`

Finally simplifying:

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