Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\frac{\text{x}^3}{(\text{x}^2+1)^3}\text{dx}$
✓

Answer

Let I $=\int\frac{\text{x}^3}{(\text{x}^2+1)^3}\text{dx}\ ....(1)$
Let $1 + x^2 = t$ then,
$d(1 + x^2) = dt$
$\Rightarrow2\text{x dx}=\text{dt}$
$\Rightarrow\text{x dx}=\frac{\text{dt}}{2}$
Puttting $1 + x^2 = t$ and $\text{x dx}=\frac{\text{dt}}{2}$ in equation (1), we get
$\text{I}=\int\frac{\text{x}^2}{\text{t}^3}\times\frac{\text{dt}}{2}$
$=\frac{1}{2}\int\frac{(\text{t}-1)}{\text{t}^3}\text{dt}\ \ [\because1+\text{x}^2=\text{t}]$
$=\frac{1}{2}\int\Big[\Big(\frac{\text{t}}{\text{t}^3}-\frac{1}{\text{t}^3}\Big)\text{dt}\Big]$
$=\frac{1}{2}\int\big(\text{t}^{-2}-\text{t}^{-3}\big)\text{dt}$
$=\frac{1}{2}\Big[-1\text{t}^{-1}-\frac{\text{t}^{-2}}{-2}\Big]+\text{C}$
$=\frac{1}{2}\Big[-\frac{1}{\text{t}}+\frac{1}{2\text{t}^2}\Big]+\text{C}$
$=-\frac{1}{2\text{t}}+\frac{1}{4\text{t}^2}+\text{C}$
$=-\frac{1}{2(1+\text{x}^2)}+\frac{1}{4(1+\text{x}^2)^2}+\text{C}$
$=\frac{-2(1+\text{x}^2)+1}{4(1+\text{x}^2)^2}+\text{C}$
$=\frac{-2-2\text{x}^2+1}{4(1+\text{x}^2)^2}+\text{C}$
$=\frac{-2\text{x}^2-1}{4(1+\text{x}^2)^2}+\text{C}$
$=-\frac{(1+2\text{x}^2)}{4(\text{x}^2+1)^2}+\text{C}$
$\therefore\text{I}=-\frac{(1+2\text{x}^2)}{4(\text{x}^2+1)^2}+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\sqrt{25-\text{x}^2}\text{ on }[-3,4]$
→
Solve each of the following L.P.P. : Maximize z = 2x + 3y subject to x – y ≥ 3, x ≥ 0, y ≥ 0
→
A pair of fair dice is thrown. Let X be the random variable which denotes the minimum of the two numbers which appear. Find the probability distribution, mean and variance of X.
→
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,$f'(x) = (x - 1)(x + 2)^2$
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{bmatrix}$
→
Evalute the following integrals:
$\int\frac{1-\cot\text{x}}{1+\cot\text{x}}\text{dx}$
→
Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{x}=\text{a}(\theta-\sin\theta),\text{y}=\text{a}(1-\cos\theta)\ \text{at}\ \theta=\frac{-\pi}{2}$
→
Solve the following differential equation:
$\text{y dx}+\Big\{\text{x}\log\Big(\frac{\text{y}}{\text{x}}\Big)\Big\}\text{dy}-2\text{x dy}=0$
→
Differentiate the following functions with respect to x:
$\text{e}^{3\text{x}}\cos2\text{x}$
→
Evaluate the following integrals:$\int\limits^\pi_0\frac{\text{x}}{1+\sin\alpha\sin\text{x}}\text{ dx}$
→