Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals2 Marks
Question
Evaluate the following integral:
$\int\frac{1}{\sqrt{\text{a}^2+\text{b}^2\text{x}^2}}\text{ dx}$
✓

Answer

$\int\frac{1}{\sqrt{\text{a}^2+\text{b}^2\text{x}^2}}\text{ dx}$
$=\int\frac{\text{dx}}{\sqrt{\text{b}^2\Big({\frac{\text{a}^2}{\text{b}^2}+\text{x}^2\Big)}}}$
$=\frac{1}{\text{b}}\int\frac{\text{dx}}{\sqrt{\text{x}^2+\big(\frac{\text{a}}{\text{b}}\big)^2}}$
$=\frac{1}{\text{b}}\log\Big|\text{x}+\sqrt{\text{x}^2+\frac{\text{a}^2}{\text{b}^2}}\Big|+\text{C}$
$=\frac{1}{\text{b}}\Big[\log\Big|\text{x}+\frac{\sqrt{\text{b}^2\text{x}^2+\text{a}^2}}{\text{b}}\Big|\Big]+\text{C}$
$=\frac{1}{\text{b}}\Big[\log\Big|\frac{\text{bx}+\sqrt{\text{b}^2\text{x}^2+\text{a}^2}}{\text{b}}\Big|\Big]+\text{C}$
$=\frac{1}{\text{b}}\Big[\log\big|\text{bx}+\sqrt{\text{b}^2\text{x}^2+\text{a}^2}\big|-\log\text{b}\Big]+\text{C}$
$=\frac{1}{\text{b}}\log\big|\text{bx}+\sqrt{\text{b}^2\text{x}^2+\text{a}^2}\big|-\frac{\log\text{b}}{\text{b}}+\text{C}$
Let $\text{C}-\frac{\log\text{b}}{\text{b}}+\text{C}'$
$=\frac{1}{\text{b}}\log\big|\text{bx}+\sqrt{\text{b}^2\text{x}^2+\text{a}^2}\big|+\text{C}'$

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 "2 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

Evaluate the following:
$\text{cosec}^{-1}\Big(-\frac{2}{\sqrt3}\Big)+2\cot^{-1}(-1)$
→
A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is $\frac{1}{7}$ and that of wife's selection is $\frac{1}{5}$. What is the probability that,
None of them will be selected?
→
Let $\text{f}:\text{R}-\Big\{-\frac{3}{5}\Big\}\rightarrow\ \text{R}$ be a function defined as $\text{f(x)}=\frac{2\text{x}}{5\text{x}+3}.$ Write f-1: Range of $\text{f}\rightarrow\ \text{R}-\Big\{-\frac{3}{5}\Big\}.$
→
Examine the consistency of the system of equations:
x + 2y = 2
2x + 3y = 3
→
If f(x) = x + 7 and g(x) = x - 7, x ∈ R, write fog (7).
→
Write the value of $\begin{vmatrix}\sin20^{\circ}&-\cos20^{\circ}\\\sin70^{\circ}&\cos70^{\circ}\end{vmatrix}$
→
If the binary operation * on the set Z is defined by a * b = a + b - 5, the find the identity element with respect to *.
→
Evaluate the following integrals:
$\int\limits^{2}_1\log_\text{e}[\text{x}]\text{dx}$
→
Find the inverse of the matrix $\begin{bmatrix} 3 & -2 \\ -7 & 5 \end{bmatrix}$.
→
If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ represent the sides of a triangle taken in order, then write the value of $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}$.
→