Step 1To solve the integral.Step 2To solve∫1cos4xdxUsing sec⁡x=1cos⁡x, we have∫1cos4xdx=∫sec4xdx=∫sec2x(tan2x+1)dx [usin⁡g,sec2x=tan2x+1]we substituteu=tan⁡x⇒dudx=sec2x⇒dx=1sec2xduThus,∫1cos4xdx=∫(u2+1)du=u33+uUndo substitution u=tan⁡x; we have∫1cos4xdx=tan3x3+tan⁡x+Cwhere C is constant of integration.

Solved: "Solve.∫1cos4xdx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Shaylee Pace

Open question

2022-08-20

Solve.

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

Answer & Explanation

Mohammad Orr

Beginner2022-08-21Added 8 answers

Step 1
To solve the integral.
Step 2
To solve

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

Using

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

, we have

\int \frac{1}{\cos^{4} x} d x = \int \sec^{4} x d x

= \int \sec^{2} x (\tan^{2} x + 1) d x [u \sin g, \sec^{2} x = \tan^{2} x + 1]

we substitute

u = \tan x

\Rightarrow \frac{d u}{d x} = \sec^{2} x

\Rightarrow d x = \frac{1}{\sec^{2} x} d u

Thus,

\int \frac{1}{\cos^{4} x} d x = \int (u^{2} + 1) d u = \frac{u^{3}}{3} + u

Undo substitution

u = \tan x

; we have

\int \frac{1}{\cos^{4} x} d x = \frac{\tan^{3} x}{3} + \tan x + C

where C is constant of integration.

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