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

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

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

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

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

Solution steps

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

Simplify

(sin (x) + 1)^{3} : (sin (x) + 1)^{2} (sin (x) + 1)

= \int cos (x) (sin (x) + 1)^{2} (sin (x) + 1) d x

Simplify

cos (x) (sin (x) + 1)^{2} (sin (x) + 1) : cos (x) (sin (x) + 1)^{3}

= \int cos (x) (sin (x) + 1)^{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 = sin (x) + 1

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

Add a constant to the solution

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

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