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

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

Question
Applications of integrals
asked 2021-02-21
Evaluate the integral using the indicated substitution.
\(\displaystyle\int{{\cos}^{{{2}}}{x}}{\sin{{x}}}{\left.{d}{x}\right.},\ {u}={\cos{{x}}}\)

Answers (1)

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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