What is the integral of (cos(ax))/(sqrt(b-asin(ax))) ?

The integral of (cos(ax))/(sqrt(b-asin(ax))) is -2/(a^2)sqrt(b-asin(ax))+C

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integral of (cos(ax))/(sqrt(b-asin(ax)))

\int \frac{cos ( a x )}{b - a sin ( a x )} d x

- \frac{2}{a ^{2}} b - a sin (a x) + C

Solution steps

\int \frac{cos ( a x )}{b - a sin ( a x )} d x

Apply u-substitution

: \int - \frac{1}{a ^{2} u} d u

= \int - \frac{1}{a ^{2} u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{a ^{2}} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= - \frac{1}{a ^{2}} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = b - a sin (a x)

= - \frac{1}{a ^{2}} \cdot 2 (b - a sin (a x))^{\frac{1}{2}}

Simplify

- \frac{1}{a ^{2}} \cdot 2 (b - a sin (a x))^{\frac{1}{2}} : - \frac{2}{a ^{2}} b - a sin (a x)

= - \frac{2}{a ^{2}} b - a sin (a x)

Add a constant to the solution

= - \frac{2}{a ^{2}} b - a sin (a x) + C

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What is the integral of (cos(ax))/(sqrt(b-asin(ax))) ?
The integral of (cos(ax))/(sqrt(b-asin(ax))) is -2/(a^2)sqrt(b-asin(ax))+C