What is the integral of (cos^7(x))sin(x) ?

The integral of (cos^7(x))sin(x) is -(cos^8(x))/8+C

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integral of (cos^7(x))sin(x)

\int (cos^{7} (x)) sin (x) d x

- \frac{cos ^{8} ( x )}{8} + C

Solution steps

\int cos^{7} (x) sin (x) d x

Apply u-substitution

: \int - u^{7} d u

= \int - u^{7} d u

Take the constant out:

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

= - \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= - \frac{u ^{8}}{8}

Substitute back

u = cos (x)

= - \frac{cos ^{8} ( x )}{8}

Add a constant to the solution

= - \frac{cos ^{8} ( x )}{8} + C

\frac{d}{d x} (cos^{2} (x) - sin^{2} (x))

y^{^{'}} + \frac{y}{x} = 2 x^{4} y^{2}

\int e^{3 x - 8} d x

\frac{d}{d x} (e^{c o s (4 x)})

(6.3) (7.8)

What is the integral of (cos^7(x))sin(x) ?
The integral of (cos^7(x))sin(x) is -(cos^8(x))/8+C