Question:
If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
Updated On: Jun 23, 2024
- $cot\,x$
- $tan\,x$
- $sec\,x$
- $cosec\,x$
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The Correct Option is A
Solution and Explanation
Given, $\int \frac{f(x)}{log (\sin x)} dx=\log [log\, sin x]+c$
On differentiating both sides, we get
$ \frac{f(x)}{log (sin x)}=\frac{1}{log sin x} \frac{d}{dx}(log \,sin x)+0$
$\Rightarrow \frac{f(x)}{log (sin \,x)}=\frac{1}{log \,sin x} \times \frac{1}{sin\, x} \times \cos x $
$\Rightarrow f(x)=cot \,x$
On differentiating both sides, we get
$ \frac{f(x)}{log (sin x)}=\frac{1}{log sin x} \frac{d}{dx}(log \,sin x)+0$
$\Rightarrow \frac{f(x)}{log (sin \,x)}=\frac{1}{log \,sin x} \times \frac{1}{sin\, x} \times \cos x $
$\Rightarrow f(x)=cot \,x$
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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.
