What is the integral of cos(x)(1+sin(x))^3 ?

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

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integral of cos(x)(1+sin(x))^3

\int cos (x) (1 + sin (x))^{3} d x

\frac{( 1 + sin ( x ) ) ^{4}}{4} + C

Solution steps

\int cos (x) (1 + sin (x))^{3} d x

Apply u-substitution

: \int u^{3} d u

= \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= \frac{u ^{4}}{4}

Substitute back

u = 1 + sin (x)

= \frac{( 1 + sin ( x ) ) ^{4}}{4}

Add a constant to the solution

= \frac{( 1 + sin ( x ) ) ^{4}}{4} + C

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