Q.Integrate the following with respect to $x$:

Because integration reverses differentiation, every derivative formula gives a corresponding standard integral. The core standard results (each carries $+c$):

• $\int 0\,dx = c$ and $\int k\,dx = kx+c$ (constant $k$).
• Power rule: $\int x^{n}\,dx = \dfrac{x^{n+1}}{n+1}+c,\ (n\ne -1)$.
• $\int \dfrac1x\,dx = \log|x|+c$ (the $n=-1$ case).
• $\int \sin x\,dx = -\cos x+c$, $\int \cos x\,dx = \sin x+c$.
• $\int \sec^2 x\,dx = \tan x+c$, $\int \operatorname{cosec}^2 x\,dx = -\cot x+c$.
• $\int \sec x\tan x\,dx = \sec x+c$, $\int \operatorname{cosec} x\cot x\,dx = -\operatorname{cosec} x+c$.
• $\int e^{x}\,dx = e^{x}+c$, $\int a^{x}\,dx = \dfrac{a^{x}}{\log a}+c$.
• $\int \dfrac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1}x+c$, $\int \dfrac{1}{1+x^2}\,dx = \tan^{-1}x+c$.

Linear-shift rule. When a constant is added/subtracted to $x$ the formula is unchanged; when $x$ is multiplied by a constant, divide by that coefficient: if $\int f(x)\,dx = g(x)+c$, then $\int f(ax+b)\,dx = \dfrac1a\,g(ax+b)+c$.