What is the integral of cos(x)(2+sin(x))^5 ?

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

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integral of cos(x)(2+sin(x))^5

\int cos (x) (2 + sin (x))^{5} d x

\frac{( 2 + sin ( x ) ) ^{6}}{6} + C

Solution steps

\int cos (x) (2 + sin (x))^{5} d x

Apply u-substitution

: \int u^{5} d u

= \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{u ^{6}}{6}

Substitute back

u = 2 + sin (x)

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

Add a constant to the solution

= \frac{( 2 + sin ( x ) ) ^{6}}{6} + C

x \to 0 lim (x^{2} + 2 x - 2)

\frac{\partial}{\partial x} (e^{x} ln (1 + y))

(x + 1) \frac{d y}{d x} + y = ln (x), y (1) = 10

\frac{\partial}{\partial x} (7 y e^{5 x})

y = ln (x), y = 0, x = 4 e

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