6.1.3 Antiderivatives Involving Inverse
         Trigonometric Functions

Example 7

Find an antiderivative for `1/(1+9t^2)`.

Solution   Since `d/(dt) text(tan)^(-1) t=1/(1+t^2),` we know by the Chain Rule that `d/(dt) text(tan)^(-1)text[(]3t text[)]=3 1/(1+text[(]3t text[)]^2)=3/(1+9t^2).` Thus an antiderivative for `1/(1+9t^2)` is `1/3 text(tan)^(-1)text[(]3t text[)]`.

Checkpoint 8

Example 8

Find an antiderivative for `1/sqrt(1-4x^2)`.

Solution   Since `d/(dx) text(sin)^(-1) x=1/sqrt(1-x^2),` we know by the Chain Rule that `d/(dx) text(sin)^(-1)text[(]2x text[)]=2/sqrt(1-text[(]2x text[)]^2) = 2/sqrt(1-4x^2).` Thus an antiderivative for `1/sqrt(1-4x^2)` is `1/2 text(sin)^(-1)text[(]2x text[)].`

Checkpoint 9

Here is a somewhat more complicated problem.

Example 9

Find an antiderivative for `1/(4+9t^2)`.

Solution   We begin by rewriting the given function:

Note that `9text[/]4=text[(]3text[/]2text[)]^2`. Now, by the Chain Rule, `d/(dt) text(tan)^(-1)(3/2 t)=3/2 1/(1+9/4 t^2).` So

Thus one antiderivative is `1/6 text(tan)^(-1)(3/2 t)`.

Checkpoint 10

Checkpoint 11