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

Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=3+(\text{x}-2)^{\frac{2}{3}}\text{ on }[1,3]$

Evaluate the following integrals:
$\int\sin^4\text{x}\cos^3\text{x}\text{ dx}$

Two cards are drawn successively without replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

$\int(5\text{x}+3)\sqrt{2\text{x}-1}\text{ dx}$

Sand is pouring from a pipe at the rate of $12\ cm^3/s.$ The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm?

Find the equation of a plane which bisects perpendicularly the line joining the points A(2, 3, 4) and B(4, 5, 8) at right angles.

Examine the continuity of f, where f is defined by
$\text{f(x)}=\begin{cases} \sin{\text{x}- \cos\text{x}}, \text{if} \ \text{x}\neq0\\-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if}\ \text{x} = 0\end{cases}$
Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29.

Reduce the equation of the following planes to intercept from and find the intercepts on the coordinate axes:
4x + 3y - 6z - 12 = 0

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x - 2y + z = -4

3x - y - 2z = 3

Let $A=\{1,2,3,4\} ; B=\{3,5,7,9\} ; C=\{7,23,47,79\}$ and $f: A \rightarrow B, g: B \rightarrow C$ be defined as $f(x)=2 x+1$ and $g(x)=x^2-$ 2. Express $(g \circ f)^{-1}$ and $f^{-1} \circ g^{-1}$ as the sets of ordered pairs and verify that $(g \circ f)^{-1}=f^{-1} \circ g^{-1}$.