What is the integral of ((sin(x))^3)/((cos(x))^2) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of ((sin(x))^3)/((cos(x))^2)

\int \frac{( sin ( x ) ) ^{3}}{( cos ( x ) ) ^{2}} d x

cos (x) + sec (x) + C

Solution steps

\int \frac{( sin ( x ) ) ^{3}}{( cos ( x ) ) ^{2}} d x

Rewrite using trig identities

= \int tan^{2} (x) sin (x) d x

Rewrite using trig identities

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

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 (x)

= - (- \frac{1}{cos ( x )}) + cos (x)

Simplify

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

= cos (x) + sec (x)

Add a constant to the solution

= cos (x) + sec (x) + C

\int - cos (x) (e^{- x}) d x

\int x e^{x} + 2 d x

\int \frac{20}{x ^{3}} d x

\int \frac{2}{3 + cos ( x )} d x

\int \frac{1}{x ^{2}} \cdot cos (\frac{1}{x}) d x

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