What is the integral of cos^{-4}(x)sin(x) ?

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

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integral of cos^{-4}(x)sin(x)

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

\frac{1}{3} sec^{3} (x) + C

Solution steps

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

Apply u-substitution

: \int - \frac{1}{u ^{4}} d u

= \int - \frac{1}{u ^{4}} d u

Take the constant out:

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

= - \int \frac{1}{u ^{4}} d u

Apply the Power Rule

: - \frac{1}{3 u ^{3}}

= - (- \frac{1}{3 u ^{3}})

Substitute back

u = cos (x)

= - (- \frac{1}{3 cos ^{3} ( x )})

Simplify

- (- \frac{1}{3 cos ^{3} ( x )}) : \frac{1}{3} sec^{3} (x)

= \frac{1}{3} sec^{3} (x)

Add a constant to the solution

= \frac{1}{3} sec^{3} (x) + C

\frac{\partial}{\partial x} (1 + x ln (x y - 5))

\frac{\partial}{\partial x} (π r^{2} h)

\int e^{(x)} d x

\frac{d}{d x} (- 3 cos (2 x))

cos (x^{4})

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