What is the integral of 1/((cos(x))^4) ?

The integral of 1/((cos(x))^4) is tan(x)+(tan^3(x))/3+C

Popular >

integral of 1/((cos(x))^4)

\int \frac{1}{( cos ( x ) ) ^{4}} d x

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

Solution steps

\int \frac{1}{( cos ( x ) ) ^{4}} d x

Rewrite using trig identities

= \int sec^{4} (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{( 1 + 2 arctan ( x ) ) ^{3}}{1 + x ^{2}} d x

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

\int 4 sin^{2} (t) d t

\int \frac{7 x + 1}{x ^{3} + 6 x ^{2} + 12 x + 8} d x

\int 3^{\frac{2}{3}} d x

What is the integral of 1/((cos(x))^4) ?
The integral of 1/((cos(x))^4) is tan(x)+(tan^3(x))/3+C