Use integration by parts to establish a reduction formula for the integral.
-$$\int \cos ^ { n } x d x$$
A) $\int \cos ^ { n } x d x = \cos ^ { n - 1 } x \sin x - ( n - 1 ) \int \sin x \cos ^ { n - 2 } x d x$
B) $\int \cos ^ { n } x d x = - \cos ^ { n - 1 } x \sin x + ( n - 1 ) \int \cos ^ { n - 2 } x d x$
C) $\int \cos ^ { n } x d x = \frac { 1 } { n } \cos ^ { n - 1 } x \sin x + \frac { n - 1 } { n } \int \cos ^ { n - 2 } x d x$
D) $\int \cos ^ { n } x d x = \frac { 1 } { n } \cos ^ { n - 1 } x \sin x - \frac { n - 1 } { n } \int \cos ^ { n - 1 } x d x$

Question

Quizplus · Accepted Answer

The correct reduction formula for $\int \cos^n x dx$ is obtained by integration by parts, letting $u = \cos^{n-1} x$ and $dv = \cos x dx$, which leads to the formula in choice C. This formula correctly reduces the power of cosine by two and includes both a direct term and an integral term, reflecting the integration by parts process.

Use Integration by Parts to Establish a Reduction Formula for the Integral