Question:
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
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
Updated On: Mar 4, 2024
- $\log _{ e } 17-\log _{ e } 18$
- $\log _e 19-\log _e 20$
- $\frac{1}{2}\left(\log _e 19-\log _e 17\right)$
- $\frac{1}{2}\left(\log _e 17-\log _e 19\right)$
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The Correct Option is D
Solution and Explanation
The correct option is (D) : \(\frac{1}{2}\left(\log _e 17-\log _e 19\right)\)
Put x2=t
∫\(\frac{dt}{(t+1)(t+3)}\)
=\(\frac{1}{2}\)∫(\(\frac{1}{t+1}\)−\(\frac{1}{t+3}\))dt
f(x)=\(\frac{1}{2}\)ln(\(\frac{x^2+1}{x^2+3}\))+C
f(3)=\(\frac{1}{2}\)(ln10−ln12)+C
⇒C=0
f(4)=\(\frac{1}{2}\)ln(\(\frac{17}{19}\))
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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.
