What is the integral of cos(x)sec^5(x) ?

The integral of cos(x)sec^5(x) is tan(x)+(tan^3(x))/3+C

Popular >

integral of cos(x)sec^5(x)

\int cos (x) sec^{5} (x) d x

tan (x) + \frac{tan ^{3} ( x )}{3} + C

Solution steps

\int cos (x) sec^{5} (x) d x

Rewrite using trig identities

= \int (1 + tan^{2} (x)) sec^{2} (x) d x

Apply u-substitution

: \int 1 + u^{2} d u

= \int 1 + u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u + \int u^{2} d u

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= u + \frac{u ^{3}}{3}

Substitute back

u = tan (x)

= tan (x) + \frac{tan ^{3} ( x )}{3}

Add a constant to the solution

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

\int \frac{x}{x - 100} d x

\int \frac{4}{( 4 x ^{2} - 1 ) ^{3}} d x

\int (y^{3} - 1) e^{x} d x

\int \frac{1}{2} x 2^{- \frac{1}{4} x} d x

\int 4 cos (9 x) d x

What is the integral of cos(x)sec^5(x) ?
The integral of cos(x)sec^5(x) is tan(x)+(tan^3(x))/3+C