What is the integral of cos^4(x)sin^1(x) ?

The integral of cos^4(x)sin^1(x) is -(cos^5(x))/5+C

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integral of cos^4(x)sin^1(x)

\int cos^{4} (x) sin^{1} (x) d x

- \frac{cos ^{5} ( x )}{5} + C

Solution steps

\int cos^{4} (x) sin^{1} (x) d x

Simplify

cos^{4} (x) : cos^{3} (x) cos (x)

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

Simplify

cos^{3} (x) cos (x) sin^{1} (x) : cos^{4} (x) sin (x)

= \int cos^{4} (x) sin (x) d x

Apply u-substitution

= \int - u^{4} d u

Take the constant out:

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

= - \int u^{4} d u

Apply the Power Rule

= - \frac{u ^{5}}{5}

Substitute back

u = cos (x)

= - \frac{cos ^{5} ( x )}{5}

Add a constant to the solution

= - \frac{cos ^{5} ( x )}{5} + C

What is the integral of cos^4(x)sin^1(x) ?
The integral of cos^4(x)sin^1(x) is -(cos^5(x))/5+C