Question Number 107162 by bobhans last updated on 09/Aug/20
Commented by Dwaipayan Shikari last updated on 09/Aug/20
$$\int_{−\mathrm{5}} ^{\mathrm{5}} \sqrt[{\mathrm{6}}]{\mathrm{5x}^{\mathrm{2}} +\mathrm{6}}\:\left(\mathrm{5x}−\mathrm{2}\right)\mathrm{dx}=\int_{−\mathrm{5}} ^{\mathrm{5}} \sqrt[{\mathrm{6}}]{\mathrm{5x}^{\mathrm{2}} +\mathrm{6}}\:\left(−\mathrm{5x}−\mathrm{2}\right)=\mathrm{I} \\ $$$$\mathrm{2I}=\int_{−\mathrm{5}} ^{\mathrm{5}} −\mathrm{4}\sqrt[{\mathrm{6}}]{\mathrm{5x}^{\mathrm{2}} +\mathrm{6}}\:\mathrm{dx} \\ $$$$\mathrm{I}=−\mathrm{2}\int_{−\mathrm{5}} ^{\mathrm{5}} \sqrt[{\mathrm{6}}]{\mathrm{5x}^{\mathrm{2}} +\mathrm{6}}\:\mathrm{dx} \\ $$
Answered by john santu last updated on 09/Aug/20
![⊵JS⊴ I=2∫_0 ^5 (1/2)((5x^2 +6))^(1/6) d(5x^2 +6) −2∫_(−5) ^5 ((5x^2 +6))^(1/6) dx I=(6/7)[ (5x^2 +6)^(7/6) ]_0 ^5 −4∫_0 ^5 ((5x^2 +6))^(1/6) dx](https://www.tinkutara.com/question/Q107165.png)
$$\:\:\:\:\:\:\trianglerighteq\mathrm{JS}\trianglelefteq \\ $$$$\mathrm{I}=\mathrm{2}\underset{\mathrm{0}} {\overset{\mathrm{5}} {\int}}\:\frac{\mathrm{1}}{\mathrm{2}}\sqrt[{\mathrm{6}}]{\mathrm{5x}^{\mathrm{2}} +\mathrm{6}}\:\mathrm{d}\left(\mathrm{5x}^{\mathrm{2}} +\mathrm{6}\right)\:−\mathrm{2}\underset{−\mathrm{5}} {\overset{\mathrm{5}} {\int}}\sqrt[{\mathrm{6}}]{\mathrm{5x}^{\mathrm{2}} +\mathrm{6}}\:\mathrm{dx} \\ $$$$\mathrm{I}=\frac{\mathrm{6}}{\mathrm{7}}\left[\:\left(\mathrm{5x}^{\mathrm{2}} +\mathrm{6}\right)^{\mathrm{7}/\mathrm{6}} \:\right]_{\mathrm{0}} ^{\mathrm{5}} \:−\mathrm{4}\underset{\mathrm{0}} {\overset{\mathrm{5}} {\int}}\sqrt[{\mathrm{6}}]{\mathrm{5x}^{\mathrm{2}} +\mathrm{6}}\:\mathrm{dx} \\ $$$$ \\ $$