Evaluate
∫
sec
3
⁡
(
𝜃
)
tan
5
⁡
(
𝜃
)
𝑑
𝜃
.

Question

Evaluate 
∫
sec
3
⁡
(
𝜃
)
tan
5
⁡
(
𝜃
)
𝑑
𝜃
.

BlackTom AI · Accepted Answer

To evaluate the integral ∫ sec^3(θ) tan^5(θ) dθ, I will work by expressing everything in terms of sec(θ) and d(sec θ). 
Option A: (1/5) sec^5(θ) − (1/3) sec^3(θ) + C. This attempt suggests integrating to lower powers of sec, but differentiating these terms would produce derivatives involving tan(θ) sec^3(θ) or similar forms, not matching the original integrand structure. In short, this pattern does not align with the substitution path we need (u-substitution with u = sec θ) and would not generate the tan^5(θ) factor appropriately.
Option B (the correct path): (1/7) sec^7(θ) − (2/5) sec^5(θ) + (1/3) sec^3(θ) + C. Start by rewriting tan^5(θ) as tan^4(θ)·tan(θ) and use tan^2(θ) = sec^2(θ) − 1 to convert everything into powers of sec(θ). Then set u = sec(θ), so du = sec(θ)tan(θ) dθ. After replacing tan^4(θ) with (sec^2(θ) − 1)^2 and simplifying, the integral becomes ∫ [u^2(u^2 − 1)^2] du, which expands to ∫ (u^6 − 2u^4 + u^2) du. Integrating yields (1/7)u^7 − (2/5)u^5 + (1/3)u^3 + C, and substituting back u = sec(θ) gives exactly the stated result in Option B.
Option C: (1/7) sec^7(θ) − (1/5) sec^5(θ) + C. While this mirrors the first term's power, the coefficient of the sec^5(θ) term should be (2/5), not (1/5), from the expansion of ∫ (u^6 − 2u^4 + u^2) du. This mismatch means Option C is incorrect.
Option D: (1/3) sec^3(θ) − (1/5) sec^5(θ) + C. This choice rearranges the terms and uses the same powers but the coefficients do not come from the correct antiderivative produced by the u-substitution; it misses the (1/7) sec^7(θ) term and misallocates coefficients, so it does not differentiate back to the original integrand.
Option E: (1/7) sec^7(θ) − (1/3) sec^3(θ) + C. Although it begins with the correct leading term, the coefficient of the sec^3(θ) term should be +1/3, but the incorrect subtraction of (1/3) sec^3(θ) without the intermediate −(2/5)sec^5(θ) term means this expression does not differentiate to sec^3 θ tan^5 θ. Thus it’s not the correct antiderivative.

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 46 (9.7,9.8 and 9.9)

Evaluate ∫ sec 3 ⁡ ( 𝜃 ) tan 5 ⁡ ( 𝜃 ) 𝑑 𝜃 .

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