What is the integral of (-sin(x))/(cos^9(x)) ?

The integral of (-sin(x))/(cos^9(x)) is -1/8 sec^8(x)+C

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integral of (-sin(x))/(cos^9(x))

\int \frac{- sin ( x )}{cos ^{9} ( x )} d x

- \frac{1}{8} sec^{8} (x) + C

Solution steps

\int \frac{- sin ( x )}{cos ^{9} ( x )} d x

Take the constant out:

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

= - \int \frac{sin ( x )}{cos ^{9} ( x )} d x

Apply u-substitution

= - \int - \frac{1}{u ^{9}} d u

Take the constant out:

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

= - (- \int \frac{1}{u ^{9}} d u)

Apply the Power Rule

= - (- (- \frac{1}{8 u ^{8}}))

Substitute back

u = cos (x)

= - (- (- \frac{1}{8 cos ^{8} ( x )}))

Simplify

- (- (- \frac{1}{8 cos ^{8} ( x )})) : - \frac{1}{8} sec^{8} (x)

= - \frac{1}{8} sec^{8} (x)

Add a constant to the solution

= - \frac{1}{8} sec^{8} (x) + C

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What is the integral of (-sin(x))/(cos^9(x)) ?
The integral of (-sin(x))/(cos^9(x)) is -1/8 sec^8(x)+C