What is the integral of sec^2(θ)sin^3(θ) ?

The integral of sec^2(θ)sin^3(θ) is cos(θ)+sec(θ)+C

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integral of sec^2(θ)sin^3(θ)

\int sec^{2} (θ) sin^{3} (θ) d θ

cos (θ) + sec (θ) + C

Solution steps

\int sec^{2} (θ) sin^{3} (θ) d θ

Simplify

sec^{2} (θ) sin^{3} (θ) : tan^{2} (θ) sin (θ)

= \int tan^{2} (θ) sin (θ) d θ

Rewrite using trig identities

= \int \frac{( 1 - cos ^{2} ( θ ) ) sin ( θ )}{cos ^{2} ( θ )} d θ

Apply u-substitution

: \int - \frac{1 - u ^{2}}{u ^{2}} d u

= \int - \frac{1 - u ^{2}}{u ^{2}} d u

Expand

- \frac{1 - u ^{2}}{u ^{2}} : - \frac{1}{u ^{2}} + 1

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

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

\int 1 d u = u

= - (- \frac{1}{u}) + u

Substitute back

u = cos (θ)

= - (- \frac{1}{cos ( θ )}) + cos (θ)

Simplify

- (- \frac{1}{cos ( θ )}) + cos (θ) : cos (θ) + sec (θ)

= cos (θ) + sec (θ)

Add a constant to the solution

= cos (θ) + sec (θ) + C

\int 50 x^{2} d x

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What is the integral of sec^2(θ)sin^3(θ) ?
The integral of sec^2(θ)sin^3(θ) is cos(θ)+sec(θ)+C