New posts in Definite Integrals

Can someone please help me explain this fallacy

Stuck on how to approach integral $\int_0^1x^n\log(x)\,dx$

Let $p=1+\frac{1}{\sqrt 2}+\cdots\frac{1}{\sqrt {120}}$ and $q=\frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots\frac{1}{\sqrt {121}}$ then

Prove $\int_0^1 \frac{dx}{(x-2) \sqrt[5]{x^2{(1-x)}^3}} = -\frac{2^{\frac{11}{10}} \pi}{\sqrt{5+\sqrt{5}}}$

Prove that $\left|30240\int_{0}^{1}x(1-x)f(x)f'(x)dx\right|\le1$.

How to evaluate $\int _0^1\frac{\ln ^2\left(1-x\right)\ln ^3\left(1+x\right)}{1+x}\:dx$

Prove: $\int_0^2 \frac{dx}{\sqrt{1+x^3}}=\frac{\Gamma\left(\frac{1}{6}\right)\Gamma\left(\frac{1}{3}\right)}{6\Gamma\left(\frac{1}{2}\right)}$

Is the summation $\sum_{i=1}^{n}\frac1{i} \binom{n}{i}$ possible?

Prove: $\int_0^{\infty} \frac{\ln{(1+x)}\arctan{(\sqrt{x})}}{4+x^2} \, \mathrm{d}x = \frac{\pi}{2} \arctan{\left(\frac{1}{2}\right)} \ln{5}$

Proving $\int_0^\infty\left(\frac{x^xe^{-x}}{\Gamma(x+1)}-\frac1{\sqrt{2\pi x}}\right)dx=-\frac13$