0-pi-2-log-sin-d- Tinku Tara June 3, 2023 Integration FacebookTweetPin Question Number 313 by Vishal Bhardwaj last updated on 25/Jan/15 ∫0π2logsinθdθ Answered by prakash jain last updated on 20/Dec/14 I=∫0π/2lnsinθdθ…..(i)putθ=π2−α,dθ=−dαI=∫π/20lnsin(π2−α)(−dα)=−∫π/20lncosαdα=∫0π/2lncosαdα=∫0π/2lncosθdθ….(ii)add(i)and(ii)2I=∫0π/2(lnsinθ+lncosθ)dθ2I=∫0π/2lnsin2θ2dθ2I=∫0π/2(lnsin2θ−ln2)dθ2I=∫0π/2lnsin2θdθ−∫0π/2ln2dθInfirstintegralput2θ=β,2dθ=dβsotheequationonenowwithupdatedlimits2I=∫0πlnsinβdβ2−∫0π/2ln2dθ2I=12∫0πlnsinβdβ−∫0π/2ln2dθ….(iii)Nowwewillevaluatefirstintegral∫0πlnsinβdβ=∫0π/2lnsinβdβ+∫π/2πlnsinβdβ∫0πlnsinβdβ==I+∫π/2πlnsinβdβputβ=π−tdβ=−dt∫0πlnsinβdβ==I+∫π/20lnsin(π−t)(−dt)∫0πlnsinβdβ==I−∫π/20lnsintdt∫0πlnsinβdβ==I+∫0π/2lnsintdt∫0πlnsinβdβ==I+I=2I….(iv)puttingthisvaluein(iii)2I=12(2I)−∫0π/2ln2dθ2I=I−π2ln2I=−π2ln2 Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: lim-x-x-2-x-1-tan-1-x-Next Next post: f-R-R-g-R-R-f-x-determinant-x-g-x-g-x-x-g-x-determinant-f-x-x-x-f-x-