Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS3 Marks
Question
Evaluate the following integrals:
$\int\frac{\text{x}^2\text{dx}}{\text{x}^6-\text{a}^6}\text{dx}$
✓

Answer

$\int\frac{\text{x}^2\text{dx}}{\text{x}^6-\text{a}^6}$
Let $\text{x}^3=\text{t}$
$\Rightarrow3\text{x}^2\text{dx = dt}$
$\Rightarrow\text{x}^2\text{dx}=\frac{\text{dt}}{3}$
Now, $\int\frac{\text{x}^2\text{dx}}{\text{x}^6-\text{a}^6}$
$=\frac{1}{3}\int\frac{\text{dt}}{\text{t}^2-(\text{a}^3)^2}$
$=\frac{1}{3}\times\frac{1}{2\text{a}^3}\log\bigg|\frac{\text{t}-\text{a}^3}{\text{t}+\text{a}^3}\Big|+\text{C}$
$=\frac{1}{6\text{a}^3}\log\Big|\frac{\text{x}^3-\text{a}^3}{\text{x}^3+\text{a}^3}\Big|+\text{C}$

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 "3 Marks" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the approximate change in the value V of a cube of side x metres caused by increasing the side by 1%.
→
If $A$ and $B$ has same order invertible square matrix, then prove that :$
(AB)^{-1}=B^{-1} \cdot A^{-1}
$
→
If $\text{y}=\cos^{-1}\text{x},$ Find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ in terms of y alone.
→
Evaluate the following:
$\int\limits^2_1\frac{\text{dx}}{\sqrt{(\text{x}-1)(2-\text{x})}}$
→
Find the values of a, b, c and d if:

$3\begin{bmatrix}\text{a}&\text{b}\\\text{c}&\text{d}\end{bmatrix}=\begin{bmatrix}\text{a}&6\\-1&2\text{d}\end{bmatrix}+\begin{bmatrix}4&\text{a}+\text{b}\\\text{c}+\text{d}&3\end{bmatrix}.$

→
Evaluate the following integrals:
$\int\big(2-3\text{x}\big)\big(3+2\text{x}\big)\big(1-2\text{x}\big)\text{dx}$
→
Can the mean of a binomial distribution be less than its variance?
→
Differentiate the following functions with respect to x:
$\tan^{-1}(\text{e}^{\text{x}})$
→
Find the probability distribution of
number of tails in the simultaneous tosses of three coins.
→
Differential equation $\frac{\text{dy}}{\text{dx}}+\text{y}=0,\text{y}(0)=3$

Function $\text{y}=\text{e}^\text{-x}+2$

→