Download App

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

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]4 Marks
Question
$\int x^3 \tan ^{-1} x d x$
✓

Answer

$\text { Let } I =\int x^3 \cdot \tan ^{-1} x \cdot d x$
$=\int\left(\tan ^{-1} x\right) x^3 d x$
$=\left(\tan ^{-1} x\right) \int x^3 d x-\int\left[\frac{ d }{ d x}\left(\tan ^{-1} x\right) \int x^3 d x\right] d x$
$=\left(\tan ^{-1} x\right) \cdot\left(\frac{x^4}{4}\right)-\int \frac{1}{1+x^2} \cdot \frac{x^4}{4} d x$
$=\frac{x^4}{4} \tan ^{-1} x+\frac{1}{4} \int \frac{\left(-x^4\right)}{1+x^2} \cdot d x$
$=\frac{x^4}{4} \tan ^{-1} x+\frac{1}{4} \int \frac{\left(1-x^4\right)-1}{1+x} d x$
$=\frac{x^4}{4} \tan ^{-1} x+\frac{1}{4} \int \frac{\left(1-x^2\right)\left(1+x^2\right)-1}{1+x^2} d x$
$=\frac{x^4}{4} \tan ^{-1} x+\frac{1}{4} \int\left(1-x^2-\frac{1}{1+x^2}\right) d x$
$=\frac{x^4}{4} \tan ^{-1} x+\frac{1}{4}\left(x-\frac{x^3}{3}-\tan ^{-1} x\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

If $y=\sin \left(m \cos ^{-1} x \right)$, then show that $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}+m^2 y=0$
→
Let $A (\bar{a})$ and $B (\bar{b})$ be any two points in the space $R (\bar{r})$ be a point on the line segment $AB$ dividing it internally in the ratio $m: n$, then prove that $\bar{r}=\frac{m \bar{b}+n \bar{a}}{m+n}$. Hence find the position vector $R$ which divides the line segment joining the points $A(1,-2,1)$ and $B(1,4,-2)$ internally in the ratio $2: 1$
→
Integrate the following w. r. t. x:

$\frac{5 x^2+20 x+6}{x^3+2 x^2+x}$

→
Find the area of the region lying between the parabolas $4y^2 = 9x$ and $3x^2 = 16y$
→
The cost of $2$ books, $6$ notebooks and $3$ pens is $₹ 40$ . The cost of $3$ books, $4$ notebooks and $2$ pens is $₹ 35$ , while the cost of $5$ books, $7$ notebooks and $4$ pens is $₹ 61$ . Using this information and matrix method, find the cost of $1$ book, $1$ notebook and $1$ pen separately.
→
Let $X$ denote the sum of the numbers obtained when two fair dice are rolled. Find the standard deviation of $X.$
→
Show the equation $2 x^2+x y-y^2+x+4 y-3=0$ represents a pair of lines. Also find the acute angle between them.
→
Find the area enclosed between the circle $x^2 + y^2 = 9,$ along $X$-axis and the line $x = y,$ lying in the first quadrant
→
Prove that: $\int_0^{2 a } f (x) d x=\int_0^{ a } f (x) d x+\int_0^{ a } f (2 a -x) d x$
→
Find the inverse of the following matrices by the adjoint method : $\left[\begin{array}{cc}2 & -2 \\ 4 & 3\end{array}\right]$
→