Question

Use the substitution u = tan x to evaluate the integral:
[ ∫ tan³ x sec x dx ]

Answer 1

To evaluate the integral ∫ tan³ x sec x dx using the substitution u = tan x, we can substitute back into the integral after calculating the values of u and du. The final result is (tan³ x)/3 + C.

Step-by-step explanation:

To evaluate the integral ∫ tan³ x sec x dx using the substitution u = tan x, we can substitute back into the integral after calculating the values of u and du.

Let u = tan x, then du = sec² x dx. Substituting these values into the integral:

∫ tan³ x sec x dx = ∫ (u³)(1/du) = ∫ u² du = u³/3 + C

Substituting back u = tan x, we have the final result: ∫ tan³ x sec x dx = (tan³ x)/3 + C.