Evaluate the following integrals: x^2+x-1 x^2+x-6 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\text{x}^2+\text{x}-1}{\text{x}^2+\text{x}-6}\text{ dx}$

✓

Answer

Let $\text{I}=\int\frac{\text{x}^2+\text{x}-1}{\text{x}^2+\text{x}-6}\text{ dx}$
$=\int\Big[1+\frac{5}{\text{x}^2+\text{x}-6}\Big]\text{dx}$
$=\text{x}+\int\frac{5}{\text{x}^2+\text{x}-6}\text{ dx}+\text{C}_1\ ....(1)$
$\text{I}_1=5\int\frac{1}{\text{x}^2+\text{x}-6}\text{ dx}$
$=5\int\frac{1}{\text{x}^2+2\text{x}\big(\frac{1}{2}\big)+\big(\frac{1}{2}\big)^2-\big(\frac{1}{2}\big)^2-6}\text{ dx}$
$=5\int\frac{1}{\big(\text{x}+\frac{1}{2}\big)^2-\big(\frac{5}{2}\big)^2}\text{ dx}$
$5\times\frac{1}{2\big(\frac{5}{2}\big)}\log\bigg|\frac{\text{x}+\frac{1}{2}-\frac{5}{2}}{\text{x}+\frac{1}{2}+\frac{5}{2}}\bigg|+\text{C}_2$
$\Big[\text{since},\int\frac{1}{\text{x}^2-\text{a}^2}\text{ dx}=\frac{1}{2\text{a}}\log\Big|\frac{\text{x}-\text{a}}{\text{x}+\text{a}}\Big|+\text{C}\Big]$
$\text{I}_1=\log\Big|\frac{\text{x}-2}{\text{x}+3}\Big|+\text{C}_2\ ....(2)$
Using equation (1) and (2)
$\text{I}=\text{x}+\log\Big|\frac{\text{x}-2}{\text{x}+3}\Big|+\text{C}$

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 "4 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove that $\Big(\frac{\text{x}}{\text{a}}\Big)^\text{n}+\Big(\frac{\text{y}}{\text{b}}\Big)^\text{n}=2$ touches the straight line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2$ for all n $\in$ N, at all the point (a, b).

Solve the following differential equation:
$(\text{x}+\text{y})^2\frac{\text{dy}}{\text{dx}} = 1$

Solve the following differential equation $(\text{x}+2\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y},$ given that when x = 2, y = 1.

Tangent to the circle $\text{x}^{2} + \text{y}^{2} = 4$ at any point on it in the first quadrant makes intercepts OA and OB on x and y axes respectively, O being the centre of the circle. Find the minimum value of (OA + OB).

Show that $\text{f}\text{(x)}=\begin{cases}\frac{\sin 3\text{x}}{\tan2\text{x}},&\text{if } \text{x}<0\\\frac{3}{2},&\text{if }\text{x} = 0\\\frac{\log(1+3\text{x})}{\text{e}^{2\text{x}}},&\text{if}\text{ x}>0\end{cases}$ is discontinuous at x = 0.

Solve the following system of equations by matrix method:

Evaluate the following integrals:
$\int(\text{x}-3)\sqrt{\text{x}^2+3\text{x}-18}\text{dx}$

A and B toss a coin alternately till one of them gets a head and wins the game. If A starts the game, find the probability that B will win the game.

If $\text{x}^\text{y}-\text{y}^\text{x}=\text{a}^\text{b},$ find $=\frac{\text{dy}}{\text{dx}}.$

If ${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {c^2}$ for some c > 0 prove that $\frac{{{{\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]}^{\frac{3}{2}}}}}{{\frac{{{d^2}y}}{{d{x^2}}}}}$ is a constant independent of a and b.