Question:
Evaluate $\int \frac{x^{2}+4}{x^{4}+16}dx.$
Evaluate $\int \frac{x^{2}+4}{x^{4}+16}dx.$
Updated On: Apr 28, 2024
- $\frac{1}{2\sqrt{2}}\tan^{-1}\left(\frac{x^{2}-4}{2x\sqrt{2}}\right)+c$
- $\frac{1}{2\sqrt{2}}\tan^{-1}\left(\frac{x^{2}-4}{2\sqrt{2}}\right)+c$
- $\frac{1}{2\sqrt{2}}\tan^{-1}\left(\frac{x^{2}-4}{x\sqrt{2}}\right)+c$
- None of the above
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Let $I=\int \frac{x^{2}+4}{x^{4}+16} d x$
$=\int \frac{1+\frac{4}{x^{2}}}{x^{2}+\frac{16}{x^{2}}}$
$d x=\int \frac{1+\frac{4}{x^{2}}}{\left(x-\frac{4}{x}\right)^{2}+8} d x$
Putting $x-\frac{4}{x}=t$
So that $\left(1+\frac{4}{x^{2}}\right) d x=d t$
$\therefore I=\int \frac{d t}{t^{2}+(2 \sqrt{2})^{2}}$
$\Rightarrow \frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{t}{2 \sqrt{2}}\right)+C$
$\Rightarrow I=\frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{x-\frac{4}{x}}{2 \sqrt{2}}\right)+C$
$=\frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{x^{2}-4}{2 x \sqrt{2}}\right)+C$
$=\int \frac{1+\frac{4}{x^{2}}}{x^{2}+\frac{16}{x^{2}}}$
$d x=\int \frac{1+\frac{4}{x^{2}}}{\left(x-\frac{4}{x}\right)^{2}+8} d x$
Putting $x-\frac{4}{x}=t$
So that $\left(1+\frac{4}{x^{2}}\right) d x=d t$
$\therefore I=\int \frac{d t}{t^{2}+(2 \sqrt{2})^{2}}$
$\Rightarrow \frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{t}{2 \sqrt{2}}\right)+C$
$\Rightarrow I=\frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{x-\frac{4}{x}}{2 \sqrt{2}}\right)+C$
$=\frac{1}{2 \sqrt{2}} \tan ^{-1}\left(\frac{x^{2}-4}{2 x \sqrt{2}}\right)+C$
Was this answer helpful?
0
0
Top Questions on Integrals of Some Particular Functions
- If \(\int \frac{dx}{(x+1)(x-2)(x-3)}=\frac{1}{k}log_e\left \{ \frac{|x-3|^3|x+1|}{(x-2)^4}\right \}+c\), then the value of k is
- WBJEE - 2023
- Mathematics
- Integrals of Some Particular Functions
- Let $f(x)=\int \frac{2 x}{\left(x^2+1\right)\left(x^2+3\right)} d x$ If $f(3)=\frac{1}{2}\left(\log _e 5-\log _e 6\right)$, then $f(4)$ is equal to
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
- If $\int \sqrt{\sec 2 x-1} d x=\alpha \log _e\left|\cos 2 x+\beta+\sqrt{\cos 2 x\left(1+\cos \frac{1}{\beta} x\right)}\right|+$ constant, then $\beta-\alpha$ is equal to_______
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
- If $\int \limits_0^\pi \frac{5^{\cos x}\left(1+\cos x \cos 3 x+\cos ^2 x+\cos ^3 x \cos 3 x\right) d x}{1+5^{\cos x}}=\frac{ k \pi}{16}$, then $k$ is equal to
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
- The integral $16 \int\limits_1^2 \frac{d x}{x^3\left(x^2+2\right)^2}$ is equal to
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
View More Questions
Questions Asked in VITEEE exam
- Two identical blocks A and B, each of mass 'm' resting on a smooth floor are connected by a light spring of natural length L and spring constant K, with the spring at its natural length. A third identical block 'C' (mass m) moving with a speed v along the line joining A and B collides with A. the maximum compression in the spring is
- VITEEE - 2023
- work, energy and power
- Assertion :
The binding energy of the nucleus increases with the increase in atomic number.
Reason :
Heavier elements have a greater number of non-radioactive isotopes than radioactive isotopes. - The half-life of radioactive radon is 3.8 days. The time at the end of which \(\frac{1}{20}th\) of the Radon sample will remain undecayed is (given log10e=0.4343)
- Which element has more electron gain enthalpy among chalcogen group?
- VITEEE - 2022
- sulphur
- Coin is tossed till heads is obtained what is expectation of no. of coin tosses
- VITEEE - 2022
- Random Variables
View More Questions
Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
