Question

Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.) 7\sin^{8}(x)\cos(x)\ln(\sin(x))dx no. 101. \int u^{n}\ln u du=\frac{u^{n+1}{(n+1)^{2}}[(n+1)\ln u-1]+C

Integrals
ANSWERED
asked 2021-08-15
Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)
\(\displaystyle{7}{{\sin}^{{{8}}}{\left({x}\right)}}{\cos{{\left({x}\right)}}}{\ln{{\left({\sin{{\left({x}\right)}}}\right)}}}{\left.{d}{x}\right.}\)
no. 101. \(\displaystyle\int{u}^{{{n}}}{\ln{{u}}}{d}{u}={\frac{{{u}^{{{n}+{1}}}{\left\lbrace{\left({n}+{1}\right)}^{{{2}}}\right\rbrace}{\left[{\left({n}+{1}\right)}{\ln{{u}}}-{1}\right]}+{C}}}{}}\)

Expert Answers (1)

2021-08-16
Step 1
Integral: \(\displaystyle=\int{7}{\left({{\sin}^{{{8}}}{x}}\right)}\cdot{\cos{{x}}}{\ln{{\left({\sin{{x}}}\right)}}}{\left.{d}{x}\right.}\)
Let us \(\displaystyle{\sin{{x}}}={u}\)
\(\displaystyle{\cos{{x}}}{\left.{d}{x}\right.}={d}{u}\)
Substituting it in integral
Integral: \(\displaystyle={7}\int{u}^{{{8}}}{\ln{{u}}}{d}{u}\)
In table of Integral it is in born of integral no. 101. \(\displaystyle\int{u}^{{{n}}}\mu{d}{u}={\frac{{{U}^{{{n}+{1}}}}}{{{\left({n}+{1}\right)}^{{{2}}}}}}{\left[{\left({n}+{1}\right)}{\ln{{u}}}-{1}\right]}+{c}\)
\(\displaystyle{n}={8}\) Integral: \(\displaystyle={7}{\left[{\frac{{{u}^{{{8}+{1}}}}}{{{\left({8}+{1}\right)}^{{{2}}}}}}{\left[{\left({8}-{11}\right)}\mu-{1}\right]}\right]}+{c}\)
\(\displaystyle={\frac{{{7}{u}^{{{9}}}}}{{{81}}}}{\left[{9}\mu-{1}\right]}+{c}\)
\(\displaystyle={7}{\frac{{{{\sin}^{{{9}}}{x}}}}{{{81}}}}{\left[{9}{m}{\left({\sin{{x}}}\right)}-{1}\right]}+{c}\)
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