Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 204437 by SANOGO last updated on 17/Feb/24

$${\int }_0^{\frac{\mathrm{\Pi }}{2}}sin\left(t\right)ln\left(sint\right)dt$$

Answered by TonyCWX08 last updated on 18/Feb/24

$$\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}ln\left(sin\left(t\right)\right)sin\left(t\right)dt$$$$letu=ln\left(sin\left(t\right)\right)\Rightarrow du=\frac{cos\left(t\right)}{sin\left(t\right)}dt$$$$dv=sin\left(t\right)\Rightarrow v=-cos\left(t\right)$$$$$$$$Usingintergrationbyparts$$$$-cos\left(t\right)ln\left(sin\left(t\right)\right)-\int -cos\left(t\right)\times \frac{cos\left(t\right)}{sin\left(t\right)}dt$$$$=-cos\left(t\right)ln\left(sin\left(t\right)\right)+\int \frac{{cos}^2\left(t\right)}{sin\left(t\right)}dt$$$$=-cos\left(t\right)ln\left(sin\left(t\right)\right)+\int \frac{1-{sin}^2\left(t\right)}{sin\left(t\right)}dt$$$$=-cos\left(t\right)ln\left(sin\left(t\right)\right)+\int \frac{1}{sin\left(t\right)}-sin\left(t\right)dt$$$$=-cos\left(t\right)ln\left(sin\left(t\right)\right)+ln\left(tan\left(\frac{x}{2}\right)\right)+cos\left(t\right)$$$$Substitutetheboundaries$$$$=(-\left(0\right)\left(0\right)+ln\left(error\right)+0)-(-1\times error+error+1)$$$$=ERROR$$$$$$$$Answer=DoesNotExist$$

Commented by Frix last updated on 18/Feb/24

$$\mathrm{You}\mathrm{have}\mathrm{to}\mathrm{use}\mathrm{the}\mathrm{limits}.\mathrm{The}\mathrm{answer}\mathrm{is}$$$$\ln 2-1$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com