Maths

Let I = ${\displaystyle {\int }_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{sinxcosx}{co{s}^{4}x+si{n}^{4}x}}dx.$

= ${\displaystyle {\int }_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{sinxcosx}{co{s}^{4}x\left(1+\frac{si{n}^{4}x}{co{s}^{4}x}\right)}}dx$

= ${\displaystyle {\int }_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{\frac{sinx}{cosx}·\frac{cosx}{cosx}\frac{1}{co{s}^{2}x}}{\left(1+ta{n}^{4}x\right)}}dx$

= ${\displaystyle {\int }_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{tanx·se{c}^{2}x}{1+ta{n}^{4}x}}dx$

= $\frac{1}{2}{\displaystyle {\int }_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{2tanx·se{c}^{2}x}{1+ta{n}^{4}x}}dx.$

Putting tan²x = t =>2 tan x?sec²xdx = dt

When x = 0, t = tan²x = tan² 0 = 0

x = $\frac{\pi }{4}$ t = tan² $\frac{\pi }{4}$ = 1² = 1.

? I = $\frac{1}{2}{\displaystyle {\int }_0^1\frac{dt}{1+t^2}}=\frac{1}{2}{\left[ta{n}^{-1}t\right]}_0^1$

= $\frac{1}{2}\left[ta{n}^{-1}(1)-ta{n}^{-1}(0)\right]$

= $\frac{1}{2}\left[\frac{\pi }{4}-0\right]=\frac{\pi }{8}$

Let I = ${\displaystyle {\int }_{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\pi }{e}^x}\left(\frac{1-sinx}{1-cosx}\right)dx.$

= ${\displaystyle {\int }_{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\pi }{e}^x}\left[\frac{1-2sin\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.cos\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{2si{n}^2\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]dx.$

= ${\displaystyle {\int }_{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\pi }{e}^x}\left[\frac{1}{2}cose{c}^2\frac{x}{2}-cot\frac{x}{2}\right]dx$

= – ${\displaystyle {\int }_{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\pi }{e}^x}\left[cot\frac{x}{2}-\frac{1}{2}cose{c}^2\frac{x}{2}\right]dx$

= $-{\left[e^x·cot\frac{x}{2}\right]}_{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\pi }$ $\{?\int e^x\left[f(x)+f^{\prime }(x)\right]dx=e^{n}f(x)$

= $-\left[e^{\pi }cot\frac{x}{2}-\frac{\pi }{e^2}cot\frac{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{2}\right]$

= $-\left[e^{\pi }*0-e^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}·cot\frac{1}{4}\right]$

= $0+e^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}*1.$

= $e^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Kindly go through the solution

Kindly go through the solution

Let $I={{\displaystyle \int }}^{\text{?}}\frac{x^2+x+1}{{(x+1)}^2(x+2)}dx$

The integrate is of form,

$\frac{x^2+x+1}{{(x+1)}^2(x+2)}=\frac{A}{x+1}+\frac{B}{{(x+1)}^2}+\frac{C}{x+2}.$

$\Rightarrow x^2+x+1=A(x+1)(x+2)+B(x+2)+C{\left(x+1\right)}^2.$

$=A\left(x^2+3x+2\right)+B(x+2)+C\left(x^2+1+2x\right)$

Comparing the co-efficients,

A + C = 1 .(1)

3A + B + C = 1 .(2)

2A + 2B + C = 1 .(3)

Equation. (2) – 2 * Equation (1),

$3A+B+2C-2A-2C=1-2*1.$

$\Rightarrow$ A + B = –1 .(4)

Equation (3) - (1),

2A + 2B + C = A – C = 1 – 1

$\Rightarrow$ A + 2B = 0 .(5)

Equation (5) - Equation (4),

A + 2B - A - B = 0 - (-1)

$\Rightarrow$ B = 1.

From (4), A = -1 - B = -1 - 1 = -2.

And from (1), C = 1 - A = 1 - (–2) = 1+2=3

$\frac{x^2+x+1}{{(x+1)}^2(x+2)}=\frac{-2}{x+1}+\frac{1}{{(x+1)}^2}+\frac{3}{x+2}$

$\therefore I={{\displaystyle \int }}^{\text{?}}\frac{-2}{x+1}dx+{{\displaystyle \int }}^{\text{?}}\frac{1}{{(x+1)}^2}dx+{{\displaystyle \int }}^{\text{?}}\frac{3}{x+2}dx$

$=-2log|x+1|+{{\displaystyle \int }}^{\text{?}}(x+1){}^{-2}dx+3log|x+2|+C.$

$=-2log|x+1|+\frac{{(x+1)}^{-2+1}}{-2+1}+3log|x+2|+c$

$=-2log|x+1|-\frac{1}{x+1}+3log|x+2|+c.$

Let I = ${{\displaystyle \int }}^{\text{?}}\frac{2+sin2x}{1+cos2x}{e}^{x}dx\cdot$

$={{\displaystyle \int }}^{\text{?}}{e}^x\left[\frac{2+2sinxcosx}{2co{s}^{2}2}\right]dx\left\{\begin{array}{ll}\therefore sin2\theta =2sin\theta cos\theta \hfill & cos2\theta =2co{s}^2\theta -1\hfill \end{array}\right\}$

$={{\displaystyle \int }}^{\text{?}}{e}^x\left[\frac{1}{co{s}^{2}x}+\frac{sinx}{cosx}\right]dx$

$={{\displaystyle \int }}^{\text{?}}{e}^x\left[tanx+se{c}^{2}x\right]dx$ is in the form

${{\displaystyle \int }}^{\text{?}}{e}^x\left[f(x)+f^{\prime }(x)\right]dx$

$f(x)=tanx$

$f^{\prime }(x)=se{c}^{2}x.$

$\therefore I=e^{x}f(x)+c=e^{x}tanx+c.$

Kindly go through the solution

Kindly go through the solution

Kindly go through the solution

Let I $={\displaystyle ?f^{?}(ax+b)·[f]{(ax+b)}^n}dx.$

$=\frac{1}{a}{\displaystyle ?[f]{(ax+b)}^n*\left[a{f}^{?}(ax+b)\right]}dx.$

Putting f (ax + b) = t.

$f^{?}(ax+b)\frac{d}{dx}(ax+b)=\frac{dt}{dx}$

af'  (ax + b) dx = dt

$?=\frac{1}{a}{\displaystyle ?t^{x}dt}$

$=\frac{1}{a}\left[\frac{t^{n+1}}{n+1}\right]$

$=\frac{1}{a(x+1)}[f]{(ax+b)}^{n+1}+\; C$