Evaluate the following intregals: 1 (x^2+1)(x^2+2) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{(\text{x}^2+1)(\text{x}^2+2)}\ \text{dx}$

✓

Answer

Let $\text{x}^2=\text{y}$
Then, $\frac{1}{(\text{y}+1)(\text{y}+2)}=\frac{\text{A}}{\text{y}+1}+\frac{\text{B}}{\text{y}+2}$
$\Rightarrow1=\text{A}(\text{y}+2)+\text{B}(\text{y}+1)=(\text{A}+\text{B})\text{y}+(2\text{A}+\text{B})$
Equating similar terms, we get,
A + B = 0, and 2A + B = 1
Solving, we get,
Thus,
$\text{I}=\int\frac{\text{dx}}{\text{x}^2+1}-\int\frac{\text{dx}}{\text{x}^2+2}$
$\text{I}=\tan^{-1}\text{x}-\frac{1}{\sqrt{2}}\tan^{-1}\frac{\text{x}}{\sqrt{2}}+\text{C}$

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 "5 Marks Questions" 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

Write the set of values of $'a'$ for which $\text{f}(\text{x})=\log_\text{a}\text{x}$ is decreasing in its domain.

Two numbers are selected at random from integers 1 through 9. If the sum is even, find the probability that both the numbers are odd.

For the curve $y = 4x^3 – 2x^5,$ find all the points at which the tangent passes through the origin.

Find the intervals in which the function $f(x) = x^3 - 12x^2 + 36x +17$ is $(a)$ increasing, $(b)$ decreasing.

Find the minimum value of 3x + 5y subject to the constraints:

$-2\text{x}+\text{y}\leq4,\text{x}+\text{y}\geq3,$ $\text{x}-2\text{y}\leq2,\text{x},\text{y}\geq0.$

Draw a rough sketch to indicate the region bounded between the curve $y^2 = 4x$ and the line $x = 3.$ Also, find the area of this region.

Show that the following system of linear equation is inconsistent:
$2x + 3y = 5$
$6x + 9y = 10$

Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y + x}\sin\Big(\frac{\text{y}}{\text{x}}\Big)=0$

A company manufactures two types of toys A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. The profit is Rs. 50 each on type A and Rs. 60 each on type B. How many toys of each type should the company manufacture in a day to maximize the profit?

Discuss the continuity and differentiability of,$\text{f(x)}=\begin{cases}(\text{x}-\text{c})\cos\Big(\frac{1}{\text{x}-\text{c}}\Big), & \text{x}\neq 0\\0, & \text{x}= 0\end{cases}$