Download App

Question Bank [2022] — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]4 Marks
Question
$\int \frac{5 e ^x}{\left( e ^x+1\right)\left( e ^{2 x}+9\right)} d x$
✓

Answer

Let $i =\int \frac{5 e ^x}{\left( e ^x+1\right)\left( e ^{2 x}+9\right)} d x$
Put $e ^{ x }= t$
$ \therefore e ^{ x } dx = dt$
$\therefore I =\int \frac{5 dt }{( t +1)\left( t ^2+9\right)} $
Let $\frac{5}{(t+1)\left(t^2+9\right)}$
$ =\frac{A}{t+1}+\frac{B t+C}{t^2+9}$
$\therefore 5=A\left(t^2+9\right)+(B t+C)(t+1)\ldots(i) $
Putting $t=-1$ in (i), we get
$ 5= A \left[(-1)^2+9\right]$
$\therefore 5=10 A$
$\therefore A =\frac{1}{2} $
Putting $t=0$ in (i), we get
$ 5=A(0+9)+(0+C)(0+1)$
$\therefore 5=9 A+C$
$\therefore 5=9\left(\frac{1}{2}\right)+C $
$\therefore C=\frac{1}{2}$
Putting $t=1$ in (i), we get
$ 5= A \left(1^2+9\right)+( B + C )(1+1)$
$\therefore 5=10 A +2 B +2 C$
$\therefore 5=10\left(\frac{1}{2}\right)+2 B +2\left(\frac{1}{2}\right)$
$\therefore-1=2 B$
$\therefore B =-\frac{1}{2}$
$\therefore \frac{5}{( t +1)\left( t ^2+9\right)}=\frac{\frac{1}{2}}{ t +1}+\frac{\frac{1}{2} t +\frac{1}{2}}{ t ^2+9}$
$\therefore I =\int\left(\frac{\frac{1}{2}}{ t +1}+\frac{\frac{-1}{2} t +\frac{1}{2}}{ t ^2+9}\right) dt$
$=\frac{1}{2}\left[\int \frac{1}{ t +1} dt -\int \frac{ t }{ t ^2+9} dt +\int \frac{1}{ t ^2+9} dt \right]$
$=\frac{1}{2}\left[\int \frac{1}{ t +1} dt -\frac{1}{2} \int \frac{2 t }{ t ^2+9} dt +\int \frac{1}{ t ^2+3^2} dt \right]$
$=\frac{1}{2}\left[\log | t + t |-\frac{1}{2} \log \left| t ^2+9\right|+\frac{1}{3} tan ^{-1}\left(\frac{ t }{3}\right)\right]+ c$
$\therefore I =\frac{1}{2} \log \left| e ^x+1\right|-\frac{1}{4} \log \left| e ^{2 x}+9\right|+\frac{1}{6} tan ^{-1}\left(\frac{ e ^x}{3}\right)+ c $

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(4 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

$\int \frac{(2 \log x+3)}{x(3 \log x+2)\left[(\log x)^2+1\right]} d x$
→
Given the p.d.f. of a continuous random r.v. $X, f(x)=\frac{x^2}{3}$, for $-1<x<2$ and $=0$ otherwise. Determine c.d.f. of $X$ and hence find $P(X<1) ; P(X<-2), P(X>0), P(1<X<2)$.
→
Solve the differential equation : $\frac{d y}{d x}+y \sec x=\tan x$.
→
$\int \frac{x^2}{\left(x^2+1\right)\left(x^2-2\right)\left(x^2+3\right)} d x$
→
Show that the lines $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ and $\frac{x}{1}=\frac{y-7}{-3}=\frac{z+7}{2}$ are coplanar. Find the equation of the plane containing them.
→
Evaluate : $\int_0^{\pi / 2} \frac{1}{1+\sqrt[3]{\tan x}} \cdot d x$
→
Solve the following differential equations : $\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y$
→
Examine whether the statement pattern

[p → (∼q ˅ r)] ↔ ∼[p → (q → r)] is a tautology, contradiction or contingency.

→
Solve the following differential equations:

$\left(x^2+y^2\right) d x-2 x y \cdot d y=0$

→
Solve the following L.P.P. by graphical method :
Maximize : $Z=10 x+25 y$
$\begin{aligned} \text { Subject to : } & 0 \leq x \leq 3, \\ 0 & \leq y \leq 3, \\ & x+y \leq 5 .\end{aligned}$
→