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How to evaluate : $$\int \frac {(x(\pi+49))^{15/7}} {\pi ^2(x^{\pi} + 7)} dx \,?$$

My teacher gave it to me as a challenge. I tried very hard, but was not able to evaluate it. Those $\pi$s confuse me a lot.

One simple thing to note is that $\frac {(\pi+49)^{15/7}}{\pi^2}$ can be taken out of the integral sign, making the term inside the integral as $\int \frac {x^{15/7}}{x^{\pi}+7}dx.$

Can anyone tell me how to proceed ?

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    Are you given any limits? – 2017-01-16
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    @N3buchadnezzar No... Just an indefinite integral... – 2017-01-16

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Well, the integration can be easily done if we make the assumption that $\boxed{\pi \approx \frac{22}{7}}$

The the given integral (without including the constant multiplier) becomes $$\int \frac {x^{15/7}}{x^{\pi}+7}dx$$ $$=\int \frac {x^{\frac{22}{7}-1}}{x^{\pi}+7}dx$$ $$\approx \int \frac {x^{\pi-1}}{x^{\pi}+7}dx$$ $$=\frac{1}{\pi}\int \frac {\pi x^{\pi-1}}{x^{\pi}+7}dx$$ $$=\frac{1}{\pi}\int \frac {d(x^{\pi}+7)}{(x^{\pi}+7)}$$ $$=\frac{1}{\pi}\ln |x^{\pi}+7|+c$$ where $c$ is the constant of integration.

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    Nice and easy...+1 – 2017-01-16
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    Shouldn't it be $\frac {ln|x^{\pi} + 7| } {\pi} + C$ in the last step... ???? – 2017-01-16
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    Yeah, thats exactly what I wrote.. – 2017-01-16
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    I meant the "|" sign... shouldn't the numerator be in modulus??? – 2017-01-16
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    Yes, I have used modulus sign .. Please check.. – 2017-01-16
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    Nowhere to be seen .... If using modulus sign is correct then please edit ur soln – 2017-01-16
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    Well, don't u see the "|" sign on either side of $x^\pi+7$?? That's the mod sign ... – 2017-01-16
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    Please do not be angry, but you had NOT used the mod sign before... but now it is visible......... – 2017-01-16
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    You can check the edit summary of my question, if I edited it or not ... but you will see that there is no such edit summary ... because it was there from the beginning ... However, you perhaps didn't refresh your page ... So the answer did not load properly.. Nothing to be angry.. – 2017-01-16