Globokim8

2021-08-15

Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)

$7{\mathrm{sin}}^{8}\left(x\right)\mathrm{cos}\left(x\right)\mathrm{ln}\left(\mathrm{sin}\left(x\right)\right)dx$

no. 101.$\int {u}^{n}\mathrm{ln}udu=\frac{{u}^{n+1}\left\{{(n+1)}^{2}\right\}[(n+1)\mathrm{ln}u-1]+C}{}$

no. 101.

unessodopunsep

Skilled2021-08-16Added 105 answers

Step 1

Integral:$=\int 7\left({\mathrm{sin}}^{8}x\right)\cdot \mathrm{cos}x\mathrm{ln}\left(\mathrm{sin}x\right)dx$

Let us$\mathrm{sin}x=u$

$\mathrm{cos}xdx=du$

Substituting it in integral

Integral:$=7\int {u}^{8}\mathrm{ln}udu$

In table of Integral it is in born of integral no. 101.$\int {u}^{n}\mu du=\frac{{U}^{n+1}}{{(n+1)}^{2}}[(n+1)\mathrm{ln}u-1]+c$

$n=8$
Integral: $=7\left[\frac{{u}^{8+1}}{{(8+1)}^{2}}[(8-11)\mu -1]\right]+c$

$=\frac{7{u}^{9}}{81}[9\mu -1]+c$

$=7\frac{{\mathrm{sin}}^{9}x}{81}[9m\left(\mathrm{sin}x\right)-1]+c$

Integral:

Let us

Substituting it in integral

Integral:

In table of Integral it is in born of integral no. 101.

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