Question

Integrate the following functions with respect to $x$:
i) $\sin 2x+\frac{1}{x+1}$
ii) $\tan (3x+1)+e^{4x+5}$
iii) $2\tan (4x+5)$
iv) $\frac{x}{\sqrt{x+2}}$
v) ${\sin }^{2}x$

Answer 1

$\left(I\right)$ $\int (sin2x+\frac{1}{x+1})dx$

$\Rightarrow \frac{cos2x}{2}+In(x+1)+c$

$\left(II\right)$ $\int (tan(3x+1)+e^{4x+5})dx$

$=\frac{-In\left|cos(3x+1)\right|}{3}+\frac{e^{4x+5}}{4}+c$

$\left(III\right)$ $\int 2tan(4x+5)dx$

$=\frac{-1}{4}\times 2In\left|cos(4x+5)\right|+c$

$\left(IV\right)$ $\int \frac{x}{\sqrt{x+2}}dx=\int \frac{x+2-2}{\sqrt{x+2}}dx$

$=\int \sqrt{x+2}dx-2\int \frac{dx}{\sqrt{x72}}$

$\Rightarrow \frac{{(x+2)}^{3/2}}{3/2}-4\sqrt{x+2}+c$

$\left(V\right)$ $\int {sin}^{2}x=\int 1-cos2xdx=x-\frac{sin2x}{2}+c$

Hence putting in it with repultion.