Trigonometric integral Evaluate the following integrals. int sin^2 0cos^5 0d0

Trigonometric integral Evaluate the following integrals. int sin^2 0cos^5 0d0

Question
Applications of integrals
asked 2021-02-19
Trigonometric integral Evaluate the following integrals.
\(\displaystyle\int{{\sin}^{{2}}{0}}{{\cos}^{{5}}{0}}{d}{0}\)

Answers (1)

2021-02-20
Step 1
Consider the provided integral,
\(\displaystyle\int{{\sin}^{{2}}{0}}{{\cos}^{{5}}{0}}{d}{0}\)
Evaluate the following integrals.
Using the trigonometric identity,
\(\displaystyle\int{{\sin}^{{2}}{\left({0}\right)}}{{\cos}^{{5}}{\left({0}\right)}}{d}{0}=\int{\left({1}-{{\sin}^{{2}}{\left({0}\right)}}\right)}^{{2}}{\cos{{\left({0}\right)}}}{d}{0}\)
Apply u-Substitution method,
\(\displaystyle{u}={\sin{{0}}}\Rightarrow{d}{u}={\cos{{0}}}{d}{0}\)
Step 2
Therefore,
\(\displaystyle\int{{\sin}^{{2}}{\left({0}\right)}}{{\cos}^{{5}}{\left({0}\right)}}{d}{0}=\int{u}^{{2}}{\left({1}-{u}^{{2}}\right)}^{{2}}{d}{u}\)
\(\displaystyle=\int{u}^{{2}}-{2}{u}^{{4}}+{u}^{{6}}{d}{u}\)
\(\displaystyle=\int{u}^{{2}}{d}{u}-\int{2}{u}^{{4}}{d}{u}+\int{u}^{{6}}{d}{u}\)
\(\displaystyle=\frac{{u}^{{3}}}{{3}}-\frac{{{2}{u}^{{5}}}}{{5}}+\frac{{{u}^{{7}}}}{{7}}+{C}\)
Substitute back,
\(\displaystyle\int{{\sin}^{{2}}{\left({0}\right)}}{{\cos}^{{5}}{\left({0}\right)}}{d}{0}=\frac{{{{\sin}^{{3}}{0}}}}{{3}}-\frac{{{2}{{\sin}^{{5}}{0}}}}{{5}}+\frac{{{{\sin}^{{7}}{0}}}}{{7}}+{C}\)
Hence.
0

Relevant Questions

asked 2021-05-11
Evaluate the integrals \(\int \cos ec^{4}0d0\)
asked 2021-05-23
Evaluate each of the following integrals.
\(\int\frac{e^{x}}{1+e^{x}}dx\)
asked 2021-05-28
Evaluate the following integral.
\(\int \frac{3x^{2}+\sqrt{x}}{\sqrt{x}}dx\)
asked 2021-05-27
Evaluate the following integral.
\(\int 2x^{3}+3x-2dx\)
asked 2021-05-23
Use a change of variables to evaluate the following integral.
\(\int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx\)
asked 2021-05-11
Evaluate the following integral: \(\int \frac{(y-3)}{y^{2}-6y+1}\)
asked 2021-05-29
Evaluate the following integral: \(\int \frac{x+3}{x-1}dx\)
asked 2021-05-30
Evaluate the following integral: \(\int\frac{vdv}{6v^{2}-1}\)
asked 2021-05-23
Find the indefinite integral \(\int \ln(\frac{x}{3})dx\) (a) using a table of integrals and (b) using the Integration by parts method.
asked 2021-06-03
Find formulas for the functions represented by the integrals.
\(\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0\)
...