What is the integral of (sin(x))/((2+cos(x))^6) ?

The integral of (sin(x))/((2+cos(x))^6) is 1/(5(2+cos(x))^5)+C

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integral of (sin(x))/((2+cos(x))^6)

\int \frac{sin ( x )}{( 2 + cos ( x ) ) ^{6}} d x

\frac{1}{5 ( 2 + cos ( x ) ) ^{5}} + C

Solution steps

\int \frac{sin ( x )}{( 2 + cos ( x ) ) ^{6}} d x

Apply u-substitution

= \int - \frac{1}{( 2 + u ) ^{6}} d u

Take the constant out:

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

= - \int \frac{1}{( 2 + u ) ^{6}} d u

Apply u-substitution

= - \int \frac{1}{v ^{6}} d v

Apply the Power Rule

= - (- \frac{1}{5 v ^{5}})

Substitute back

= - (- \frac{1}{5 ( 2 + cos ( x ) ) ^{5}})

Simplify

= \frac{1}{5 ( 2 + cos ( x ) ) ^{5}}

Add a constant to the solution

= \frac{1}{5 ( 2 + cos ( x ) ) ^{5}} + C

What is the integral of (sin(x))/((2+cos(x))^6) ?
The integral of (sin(x))/((2+cos(x))^6) is 1/(5(2+cos(x))^5)+C