What is the integral of (sin(θ))/(cos^4(θ)) ?

The integral of (sin(θ))/(cos^4(θ)) is 1/3 sec^3(θ)+C

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integral of (sin(θ))/(cos^4(θ))

\int \frac{sin ( θ )}{cos ^{4} ( θ )} d θ

\frac{1}{3} sec^{3} (θ) + C

Solution steps

\int \frac{sin ( θ )}{cos ^{4} ( θ )} d θ

Rewrite using trig identities

= \int \frac{tan ( θ )}{cos ^{3} ( θ )} d θ

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{3} (θ) tan (θ) d θ

Apply u-substitution

: \int \frac{1}{3} d u

= \int \frac{1}{3} d u

Integral of a constant:

\int a d x = a x

= \frac{1}{3} u

Substitute back

u = sec^{3} (θ)

= \frac{1}{3} sec^{3} (θ)

Add a constant to the solution

= \frac{1}{3} sec^{3} (θ) + C

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What is the integral of (sin(θ))/(cos^4(θ)) ?
The integral of (sin(θ))/(cos^4(θ)) is 1/3 sec^3(θ)+C