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 ?
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.