Studdy AI Logo
Math  /  Calculus

Question∫tan⁡8x⋅sec⁡4xdx\int \tan ^{8} x \cdot \sec ^{4} x d x

Studdy Solution

STEP 1

1. We are dealing with an integral involving powers of tangent and secant functions.
2. The integral can be simplified using trigonometric identities and substitution.

STEP 2

1. Simplify the integral using trigonometric identities.
2. Use substitution to solve the integral.
3. Integrate and back-substitute to find the solution.

STEP 3

First, use the identity sec⁡2x=1+tan⁡2x\sec^2 x = 1 + \tan^2 x to express sec⁡4x\sec^4 x in terms of tan⁡x\tan x:
sec⁡4x=(sec⁡2x)2=(1+tan⁡2x)2\sec^4 x = (\sec^2 x)^2 = (1 + \tan^2 x)^2
Thus, the integral becomes:
∫tan⁡8x⋅(1+tan⁡2x)2 dx\int \tan^8 x \cdot (1 + \tan^2 x)^2 \, dx

STEP 4

Let u=tan⁡x u = \tan x , which implies du=sec⁡2x dx du = \sec^2 x \, dx . We need to express the integral in terms of u u :
The integral becomes:
∫u8⋅(1+u2)2⋅dusec⁡2x\int u^8 \cdot (1 + u^2)^2 \cdot \frac{du}{\sec^2 x}
Since sec⁡2x=1+tan⁡2x=1+u2\sec^2 x = 1 + \tan^2 x = 1 + u^2, we have:
∫u8⋅(1+u2) du\int u^8 \cdot (1 + u^2) \, du

STEP 5

Expand and integrate the expression:
∫u8⋅(1+u2) du=∫(u8+u10) du\int u^8 \cdot (1 + u^2) \, du = \int (u^8 + u^{10}) \, du
Integrate term by term:
=∫u8 du+∫u10 du= \int u^8 \, du + \int u^{10} \, du
=u99+u1111+C= \frac{u^9}{9} + \frac{u^{11}}{11} + C

STEP 6

Back-substitute u=tan⁡x u = \tan x to express the solution in terms of x x :
=(tan⁡x)99+(tan⁡x)1111+C= \frac{(\tan x)^9}{9} + \frac{(\tan x)^{11}}{11} + C
The solution to the integral is:
tan⁡9x9+tan⁡11x11+C\boxed{\frac{\tan^9 x}{9} + \frac{\tan^{11} x}{11} + C}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord