# Evaluate the integral using the indicated substitution. \int\cos^{2}x \sin xdx,\ u=\cos x

Question
Applications of integrals
Evaluate the integral using the indicated substitution.
$$\displaystyle\int{{\cos}^{{{2}}}{x}}{\sin{{x}}}{\left.{d}{x}\right.},\ {u}={\cos{{x}}}$$

2021-02-22
Step 1
Consider the integrals,
$$\displaystyle\int{{\cos}^{{{2}}}{x}}\cdot{\sin{{x}}}{\left.{d}{x}\right.}$$...(1)
Let, $$\displaystyle{u}={\cos{{x}}}$$
$$\displaystyle{d}{u}=-{\sin{{x}}}{\left.{d}{x}\right.}$$
Step 2
Substitute all value in equation (1) then,
$$\displaystyle\int{{\cos}^{{{2}}}{x}}\cdot{\sin{{x}}}\cdot{\left.{d}{x}\right.}=\int{u}^{{{2}}}{\left(-{d}{u}\right)}$$
$$\displaystyle=-\int{u}^{{{2}}}{d}{u}$$
$$\displaystyle=-{\frac{{{u}^{{{3}}}}}{{{3}}}}+{C}$$
Substitute back, $$\displaystyle{u}={\cos{{x}}}$$
$$\displaystyle\int{{\cos}^{{{2}}}{x}}\cdot{\sin{{x}}}\cdot{\left.{d}{x}\right.}=-{\frac{{{1}}}{{{3}}}}{{\cos}^{{{3}}}{x}}+{C}$$

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