which of the following is equivalent to ∫cos(4x)sin⁵(4x)dx? a 1/4 ∫cos udu·1/4 ∫sin⁵udu, where u = 4x b…

which of the following is equivalent to ∫cos(4x)sin⁵(4x)dx? a 1/4 ∫cos udu·1/4 ∫sin⁵udu, where u = 4x b ∫cos(4x)dx·∫sin⁵(4x)dx c 1/4 ∫u⁵du, where u = sin(4x) d ∫u⁵du, where u = sin(4x)

which of the following is equivalent to ∫cos(4x)sin⁵(4x)dx? a 1/4 ∫cos udu·1/4 ∫sin⁵udu, where u = 4x b ∫cos(4x)dx·∫sin⁵(4x)dx c 1/4 ∫u⁵du, where u = sin(4x) d ∫u⁵du, where u = sin(4x)

Answer

Explanation:

Step1: Use substitution method

Let $u = \sin(4x)$. Then $du=4\cos(4x)dx$, so $\cos(4x)dx=\frac{1}{4}du$. The original integral $\int\cos(4x)\sin^{5}(4x)dx$ becomes $\int u^{5}\cdot\frac{1}{4}du=\frac{1}{4}\int u^{5}du$.

Answer:

C. $\frac{1}{4}\int u^{5}du$, where $u = \sin(4x)$