Evaluate the following integral: 1 a^2+b^2x^2 dx - Vidyadip

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 Questions" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

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?

If f : A → A, g : A → A are two bijections, then prove that:
fog is an injection.

In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\text{k}(\text{x}^2+2),&\text{if x}\leq0\\3\text{x}+1,&\text{if x}>0\end{cases}$

Write the coefficient a, b, c of which the value of the integral $\int\limits^3_{-3}(\text{ax}^2+\text{bx}+\text{c})\text{dx}$ is independent.

The following defines a relation on N:
$\text{x} +\text{y} = 10,\text{x, y}\in\text{N}$ $$
Determine which of the above relations are reflexive, symmetric and transitive.

If $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2+\big(\vec{\text{a}}.\vec{\text{b}}\big)^2=144$ and $|\vec{\text{a}}|=4,$ find $\big|\vec{\text{b}}\big|.$

Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.

If $\text{A}=\begin{bmatrix}2&1&4\\4&1&5 \end{bmatrix}$ and $\text{B}=\begin{bmatrix}3&-1\\2&2\\1&3\end{bmatrix}.$ Write the order of $AB$ and $BA.$

Evaluate the following determinant:
$\begin{vmatrix}102&18&36\\1&3&4\\17&3&6\end{vmatrix}$

Let $'o\ '$ be a binary operation on the set $Q_0$ of all non$-$zero rational numbers defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{2}$ for all $\text{a},\text{b}\in\text{Q}_0.$ Show that $'o\ '$ is both commutative and associate.