9–61. Trigonometric integrals Evaluate the following integrals.
45. ∫ sec²x tan¹ᐟ²x dx

Guida verificata passo dopo passo

Step 1: Recognize the integral involves trigonometric functions sec²x and tan¹ᐟ²x. Recall that sec²x is the derivative of tan(x), which suggests a substitution method might be useful.

Step 2: Let u = tan(x). Then, du = sec²x dx. This substitution simplifies the integral by replacing sec²x dx with du and tan¹ᐟ²x with u¹ᐟ².

Step 3: Rewrite the integral in terms of u: ∫ sec²x tan¹ᐟ²x dx becomes ∫ u¹ᐟ² du.

Step 4: Apply the power rule for integration: ∫ uⁿ du = (uⁿ⁺¹)/(n+1), where n ≠ -1. Here, n = 1/2, so the integral becomes (u³ᐟ²)/(3/2) + C.

Step 5: Substitute back u = tan(x) to express the result in terms of x: (tan³ᐟ²x)/(3/2) + C.

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