Evaluate the following integrals: x x^4+a^4 dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{\text{x}}{\sqrt{\text{x}^4+\text{a}^4}}\text{ dx}$

✓

Answer

$\int\frac{\text{x}\text{ dx}}{\sqrt{\text{x}^4+\text{a}^4}}$ $\int\frac{\text{x}\text{ dx}}{\sqrt{(\text{x}^2)^2+(\text{a}^2)^2}}$ Let $\text{x}^2=\text{t}$ $\Rightarrow2\text{x}\text{ dx}=\text{dt}$ $\Rightarrow\text{x}\text{ dx}=\frac{\text{dt}}{2}$Now, $\int\frac{\text{x}\text{ dx}}{\sqrt{(\text{x}^2)^2+(\text{a}^2)^2}}$
$=\frac{1}{2}\int\frac{\text{x}\text{ dx}}{\sqrt{{\text{t}^2+(\text{a}^2)^2}}}$
$=\frac{1}{2}\log\Big|\text{t}+\sqrt{\text{t}^2+\text{a}^4}\Big|+\text{C}$
$=\frac{1}{2}\log\Big|\text{x}^2+\sqrt{\text{x}^4+\text{a}^4}\Big|+\text{C}$

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 Question" 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

Evaluate the following definite integrals:
$\int_{0}^\limits{1}\frac{1}{2\text{x}^2+\text{x}+1}\text{ dx}$

Write the equation of the plane whose intercepts on the coordinate axes are 2, -3 and 4

In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\text{k}+1,&\text{if}\text{ x}\leq5\\3\text{x}-5,&\text{if}\text{ x}>5\end{cases}\text{at x} =5$

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}(\text{x}-\text{a}){\sin}\Big(\frac{1}{\text{x}-\text{a}}\Big) & \text{x} \neq \text{a}\\\ 0, & \text{ x} = \text{a}\end{cases}\text{at x}=\text{a}$

Find the equation of the plane passing through the following points:
(2, 1, 0), (3, -2, -2) and (3, 1, 7)

By using the properties of definite integral, evaluate the integral in Exercise:
$\int^{\frac{\pi}{2}}_{0}\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}\text{dx}$

Find the mean and standard deviation of the following probability distributions:

$x_i$	2	3	4
$p_i$	0.2	0.5	0.3

For each of the differential equations in find the general solution:
$\frac{\text{dy}}{\text{dx}}=\text{sin}^{-1}\text{x}$

Find the simplified form of $\cos^{-1}\Big(\frac{3}{5}\cos\text{x}+\frac{4}{5}\sin\text{x}\Big),$ where $\text{x}\in\Big[\frac{-3\pi}{4},\frac{\pi}{4}\Big].$

Find values of a and b if A = B, where:
$\text{A}=\begin{bmatrix}\text{a}+4&3\text{b}\\8&-6\end{bmatrix}$ and $\text{B}=\begin{bmatrix}2\text{a}+2&\text{b}^2+2\\8&\text{b}^2-5\text{b}\end{bmatrix}$