Evalute : 3 x-1 2 x^2-x-1 d x

Question

Evalute : $\int \frac{3 x-1}{2 x^2-x-1} d x$

✓

Answer

$
\text { Let } \begin{aligned}
I & =\int \frac{3 x-1}{2 x^2-x-1} d x \\
& =\int \frac{3 x-1}{(x-1)(2 x+1)} d x
\end{aligned}
$
Let $\frac{3 x-1}{(x-1)(2 x+1)}=\frac{A}{x-1}+\frac{B}{2 x+1}$
$
\therefore 3 x-1=A(2 x+1)+B(x-1)
$
Put $x-1=0$, i.e. $x=1$, we get
$
\begin{aligned}
& 3(1)-1=A(3)+B(0) \quad \therefore 2=3 A \quad \therefore A=\frac{2}{3} \\
\end{aligned}
$
Put $2 x+1=0$, i.e. $x=-\frac{1}{2}$, we get
$
\begin{aligned}
& 3\left(-\frac{1}{2}\right)-1=A(0)+B\left(-\frac{3}{2}\right) \\
& \therefore-\frac{5}{2}=-\frac{3}{2} B \quad \therefore B=\frac{5}{3} \\
& \therefore \frac{3 x-1}{(x-1)(2 x+1)}=\frac{\left(\frac{2}{3}\right)}{x-1}+\frac{\left(\frac{5}{3}\right)}{2 x+1} \\
& \therefore I=\int\left[\frac{\left(\frac{2}{3}\right)}{x-1}+\frac{\left(\frac{5}{3}\right)}{2 x+1}\right] d x \\
& \quad=\frac{2}{3} \int \frac{1}{x-1} d x+\frac{5}{3} \int \frac{1}{2 x+1} d x \\
& \quad=\frac{2}{3} \log |x-1|+\frac{5}{3} \cdot \frac{\log |2 x+1|}{2}+c \\
& \quad=\frac{2}{3} \log |x-1|+\frac{5}{6} \log |2 x+1|+c .
\end{aligned}
$

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.(5 Marks)" from Integration (p-1)→Integration (p-1) — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

Calculate Walsh’s Price Index Number.

Commodity	Base Year		Current Year
	Price	Quantity	Price	Quantity
I	10	16	20	9
II	20	2	25	8
III	30	3	40	27
IV	60	9	75	36

→

A firm manufactures two products $A$ and $B$ on which profit is earned per unit $₹ 3 /-$ and $₹ 4 /$ - respectively. Each product is processed on two machines $M _1$ and $M _2$. Product A requires one minute of processing time on Mx and two minutes of processing time on $M _2$. B requires one minute of processing time on $M _1$ and one minute processing time on $M_2$ Machine $M_1$ is available for use for $450$ minutes while $M_2$ is available for $600$ minutes during any working day. Final the number of units of products $A$ and $B$ to be manufactured to get the maximum profit.

→

In each of the following examples, verify that the given function is a solution of the corresponding differential equation:

	Solution	D.E.
(i)	xy=log y+k	y^(')(1-xy)=y^(2)
(ii)	y=x^(n)	x^(2)(d^(2)y)/(dx^(2))-nx(dy)/(dx)+ny=0
(iii)	y=e^(x)	(dy)/(dx)=y
(iv)	y=1-log x	x^(2)(d^(2)y)/(dx^(2))=1
(v)	y=ae^(x)+be^(-x)	(d^(2)y)/(dx^(2))=y
(vi)	ax^(2)+by^(2)=5	xy(d^(2)y)/(dx^(2))+x((dy)/(dx))^(2)=y*(dy)/(dx)