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

Question

Quizplus · Accepted Answer

The correct reduction formula for the integral of $\sin^n x$ is obtained by using integration by parts, where one part is differentiated and the other is integrated. The formula correctly reduces the power of sine by two and includes both a direct term and an integral term, with coefficients that correctly account for the integration process.

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