Integrate. \int\cos^{5}(x)\sin(x)dx

Question:

Integrate.

{eq}\int\cos^{5}(x)\sin(x)dx {/eq}

Indefinite Integral in Calculus:

Sine and Cosine are two main trigonometric functions. We can use integration rules and techniques to compute the indefinite integration of a trigonometric function.

We may apply u-substitution to convert the integral into a standard form. The integrand after substitution can be directly integrated by using integration formulas.

Given: {eq}\displaystyle \int \cos^5 x \sin x dx {/eq}

Apply u-substitution {eq}u= \cos x \rightarrow \ du =- \sin x \ dx {/eq}

{eq}=\displaystyle \int -u^5 \ du {/eq}

Apply integral power rule:

{eq}=\displaystyle \dfrac{-u^6}{6} + C \quad \text{(Where C is an arbitrary constant )} {/eq}

Substitute back {eq}u= \cos x {/eq}

{eq}=\displaystyle -\dfrac{\cos^6(x)}{6} +C {/eq}

Therefore the solution is: {eq}\displaystyle {\bf{ \int \cos^5 x \sin x dx= -\dfrac{\cos^6(x)}{6} + C . }} {/eq}