Solve the integral: integral 5 × 4 x dx

Question

ScanSolve AI Tutor · Accepted Answer

To solve the integral ∫ tan⁵ x sec⁴ x dx: Step 1: Rewrite the integrand using trigonometric identities. We know that sec² x = 1 + tan² x. We can split sec⁴ x into sec² x · sec² x. ∫ tan⁵ x sec⁴ x dx = ∫ tan⁵ x sec² x sec² x dx; Substitute sec² x = 1 + tan² x for one of the sec² x terms. ∫ tan⁵ x (1 + tan² x) sec² x dx Step 2: Perform a substitution. Let u = tan x. Then, the differential du = sec² x dx. Step 3: Substitute u and du into the integral. ∫ u⁵ (1 + u²) du Step 4: Expand the integrand. ∫ (u⁵ + u⁷) du Step 5: Integrate term by term using the power rule ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C. (u⁵⁺¹)/(5+1) + (u⁷⁺¹)/(7+1) + C (u⁶)/(6) + (u⁸)/(8) + C Step 6: Substitute back u = tan x. (tan⁶ x)/(6) + (tan⁸ x)/(8) + C The final answer is (tan⁶ x)/(6) + (tan⁸ x)/(8) + C.

Solve the integral: integral 5 x 4 x dx

Answer

Step-by-step solution

Rewrite the integrand using trigonometric identities.

Perform a substitution.

Substitute $u$ and $du$ into the integral.

Expand the integrand.

Integrate term by term using the power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ .

Substitute back $u = \tan x$ .

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