Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f(x) = x^2 + \int_0^x e^{...

If $I_n={\int }_0^{\pi /4}{\tan }^{n}xdx$ (for $n>1$), then $I_n+I_{n-2}$ equals:

The value of ${\lim }_{n\to \infty }{\sum }_{r=1}^n\frac{1}{\sqrt{4n^2-r^2}}$ is:

Evaluate $\int e^x(\frac{1+x\ln x}{x})dx$ (Assume $x>0$ and base $e$ for $\ln$).

Let $f(x)$ be a continuous function satisfying $f(x)f(a-x)=1$. Then $\int_0^a \frac{1}{1 + f(x)} d...

The value of ${\int }_0^{\pi /2}\frac{{\sin }^{2025}x}{{\sin }^{2025}x+{\cos }^{2025}x}dx$ is: