$\int_1^2 \frac{x \, dx}{(x+1)(x+2)} =$

If $\int \frac{(\sqrt{x})^5}{(\sqrt{x})^7+x^6} dx = a\log\left|\frac{x^k}{1+x^k}\right| + C$; Where C is an arbitrary constant, then the value of $a.k$ is:

Match the rational functions with the correct form of the partial fractions.

[ $p,q,r,a,b,c,A,B,C$ are arbitrary constants]

Match List-I with List-II

Choose the correct answer from the options given below:

$\int \frac{(x-3)e^x}{(x-1)^3} dx$ is equal to

If $\displaystyle\int_0^3 (8x^3+5x^2-2x-k)\,dx = 3$, where $k$ is constant, then $k$ is equal to

The value of $\int_{-\pi/2}^{\pi/2}(\sin|x|+\cos|x|)dx$ equals:

Which of the following statements are TRUE?

A. $\displaystyle\int_4^9 \dfrac{1}{\sqrt{x}}\, dx = 2$

B. $\displaystyle\int_{-4}^4 (x^{101} + x^{201} + x^{301})\, dx = 4\left(\dfrac{1}{102} + \dfrac{1}{202} + \dfrac{1}{302}\right)$

C. $\displaystyle\int_{-1}^1 x^{1012}\, dx = \dfrac{2}{1013}$

D. $\displaystyle\int (x^2 + 2x + 1)\, dx = \dfrac{(x+1)^3}{3} + c$ where $c$ is an arbitrary constant

Choose the correct answer from the options given below: