∫
sec
2
⁡
𝑥
tan
7
⁡
𝑥
𝑑
𝑥
=
_
_
_
_
_
_
_
_
_
_
_

Question

BlackTom AI · Accepted Answer

We start by restating the given problem and noting the missing answer choices. The task shows the integral: ∫ sec^2(x) · tan^7(x) dx, and the provided (partial) answer appears as 1/8 tan^8(x) + C, but no explicit options are listed to evaluate against.

Step 1: Recognize a standard substitution pattern. When you have an integrand that is a product of tan^n(x) and sec^2(x), a common substitution is u = tan(x) because du = sec^2(x) dx. This transforms the integral into a simple power integral in u.

Step 2: Apply the substitution. Let u = tan(x). Then du = sec^2(x) dx, so the integral becomes ∫ u^7 du.

Step 3: Integrate in terms of u. The integral ∫ u^7 du equals u^8/8 + C.

Step 4: Substitute back to x. Replacing u with tan(x) gives (tan^8(x))/8 + C.

Step 5: Compare with the presented answer. The expression (1/8) tan^8(x) + C matches the standard result obtained from the substitution, confirming consistency with the implied answer.

Additional notes: If there were alternative answer choices, they would need to reflect the correct power of tan and the factor 1/8 arising from integrating u^7. Any option that suggests a different power of tan, a different coefficient, or introduces extra functions would be incorrect. Since no explicit options are provided here, the derivation demonstrates the correct antiderivative given the integrand structure.

∫ sec 2 ⁡ 𝑥 tan 7 ⁡ 𝑥 𝑑 𝑥 = _ _ _ _ _ _ _ _ _ _ _

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