What is the integral of (2cos(x))/((10+sin(x))^7) ?

The integral of (2cos(x))/((10+sin(x))^7) is -1/(3(sin(x)+10)^6)+C

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integral of (2cos(x))/((10+sin(x))^7)

\int \frac{2 cos ( x )}{( 10 + sin ( x ) ) ^{7}} d x

- \frac{1}{3 ( sin ( x ) + 10 ) ^{6}} + C

Solution steps

\int \frac{2 cos ( x )}{( 10 + sin ( x ) ) ^{7}} d x

Take the constant out:

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

= 2 \cdot \int \frac{cos ( x )}{( 10 + sin ( x ) ) ^{7}} d x

Apply u-substitution

: \int \frac{1}{( 10 + u ) ^{7}} d u

= 2 \cdot \int \frac{1}{( 10 + u ) ^{7}} d u

Apply u-substitution

: \int \frac{1}{v ^{7}} d v

= 2 \cdot \int \frac{1}{v ^{7}} d v

Apply the Power Rule

: - \frac{1}{6 v ^{6}}

= 2 (- \frac{1}{6 v ^{6}})

Substitute back

= 2 (- \frac{1}{6 ( 10 + sin ( x ) ) ^{6}})

Simplify

2 (- \frac{1}{6 ( 10 + sin ( x ) ) ^{6}}) : - \frac{1}{3 ( sin ( x ) + 10 ) ^{6}}

= - \frac{1}{3 ( sin ( x ) + 10 ) ^{6}}

Add a constant to the solution

= - \frac{1}{3 ( sin ( x ) + 10 ) ^{6}} + C

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What is the integral of (2cos(x))/((10+sin(x))^7) ?
The integral of (2cos(x))/((10+sin(x))^7) is -1/(3(sin(x)+10)^6)+C