Definite Integration

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JEE Main 2025 (Online) 22nd January Morning ShiftQuestion 378

Let for $f(x)=7\tan^{8}x+7\tan^{6}x-3\tan^{4}x-3\tan^{2}x$, $I_1=\displaystyle\int_{0}^{\pi/4}f(x)\,dx$ and $I_2=\displaystyle\int_{0}^{\pi/4}xf(x)\,dx$. Then $7I_1+12I_2$ is equal to :

$2\pi$$1$$\pi$$2$

JEE Main 2025 (Online) 23rd January Morning ShiftQuestion 379

The value of $\displaystyle\int_{e^{2}}^{e^{4}}\dfrac{1}{x}\left(\dfrac{e^{\left((\log_e x)^{2}+1\right)^{-1}}}{e^{\left((\log_e x)^{2}+1\right)^{-1}}+e^{\left((6-\log_e x)^{2}+1\right)^{-1}}}\right)dx$ is

Definite Integration

$1$$\log_e2$$e^{2}$$2$

JEE Main 2025 (Online) 28th January Evening ShiftQuestion 380

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\displaystyle\int_{0}^{x}tf(t)\,dt$. If $g\left(x^{3}\right)=x^{6}+x^{7}$, then value of $\displaystyle\sum_{r=1}^{15}f\left(r^{3}\right)$ is :

Definite Integration

$270$$340$$310$$320$

JEE Main 2025 (Online) 28th January Evening ShiftQuestion 381

Let $f:\mathbf{R}\to\mathbf{R}$ be a twice differentiable function such that $f(2)=1$. If $F(x)=xf(x)$ for all $x\in\mathbf{R}$, $\displaystyle\int_{0}^{2}xF'(x)\,dx=6$ and $\displaystyle\int_{0}^{2}x^{2}F''(x)\,dx=40$, then $F'(2)+\displaystyle\int_{0}^{2}F(x)\,dx$ is equal to :

Definite Integration

$13$$11$$9$$15$

AIEEE 2002 (Offline)Question 382

$\lim\limits_{n\to\infty}\frac{1^{p}+2^{p}+3^{p}+\cdots+n^{p}}{n^{p+1}}$ is

Definite Integration

$\frac{1}{p+1}$$\frac{1}{1-p}$$\frac1p-\frac{1}{p-1}$$\frac{1}{p+2}$