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Question Number 79612 by john santu last updated on 26/Jan/20

$$\int \frac{\mathrm{dx}}{\sqrt{\mathrm{x}}{(\sqrt[4]{\mathrm{x}}+1)}^{10}}=?$$

Answered by MJS last updated on 26/Jan/20

$$\int \frac{dx}{x^{\frac{1}{2}}{(x^{\frac{1}{4}}+1)}^{10}}=$$$$[t=x^{\frac{1}{4}}\to dx=4x^{\frac{3}{4}}dt]$$$$=4\int \frac{t}{{(t+1)}^{10}}dt=4\int \frac{dt}{{(t+1)}^9}-4\int \frac{dt}{{(t+1)}^{10}}=$$$$=-\frac{1}{2{(t+1)}^8}+\frac{4}{9{(t+1)}^9}=-\frac{9t+1}{18{(t+1)}^9}=$$$$=-\frac{9x^{\frac{1}{4}}+1}{18{(x^{\frac{1}{4}}+1)}^9}+C$$

Commented by peter frank last updated on 26/Jan/20

$$thanks$$

Commented by john santu last updated on 27/Jan/20

$$\mathrm{thank}\mathrm{you}\mathrm{mister}$$

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