What is the integral of (sin(x))/((9+cos(x))^4) ?

The integral of (sin(x))/((9+cos(x))^4) is 1/(3(9+cos(x))^3)+C

integral of (sin(x))/((9+cos(x))^4)

\int \frac{sin ( x )}{( 9 + cos ( x ) ) ^{4}} d x

\frac{1}{3 ( 9 + cos ( x ) ) ^{3}} + C

Solution steps

\int \frac{sin ( x )}{( 9 + cos ( x ) ) ^{4}} d x

Apply u-substitution

= \int - \frac{1}{( 9 + u ) ^{4}} d u

Take the constant out:

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

= - \int \frac{1}{( 9 + u ) ^{4}} d u

Apply u-substitution

= - \int \frac{1}{v ^{4}} d v

Apply the Power Rule

= - (- \frac{1}{3 v ^{3}})

Substitute back

= - (- \frac{1}{3 ( 9 + cos ( x ) ) ^{3}})

Simplify

= \frac{1}{3 ( 9 + cos ( x ) ) ^{3}}

Add a constant to the solution

= \frac{1}{3 ( 9 + cos ( x ) ) ^{3}} + C

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What is the integral of (sin(x))/((9+cos(x))^4) ?
The integral of (sin(x))/((9+cos(x))^4) is 1/(3(9+cos(x))^3)+C