Question:
If $f\left(x\right) = \int\limits^{sin\,x}_{2x}cos\left(t^{3}\right)dt$, then $f'{x}$ is equal to
If $f\left(x\right) = \int\limits^{sin\,x}_{2x}cos\left(t^{3}\right)dt$, then $f'{x}$ is equal to
Updated On: Apr 8, 2024
- $cos(sin\, x)cos\,x-2cos(8x^3)$
- $sin(sin^3\, x)sin -2sin(8x^3)$
- $cos(cos^3\, x)cos\,x-2\,cos(x^3)$
- $cos(sin^3\, x)-cos(8x^3)$
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The Correct Option is A
Solution and Explanation
Given, $f(x)=\int_{2 x}^{\sin x} \cos t^{3} d t$
Using Leibnitz's rule
$f^{\prime}(x)=\cos \left(\sin ^{3} x\right) \frac{d}{d x}(\sin x)$
$-\cos (2 x)^{3} \frac{d}{d x}(2 x)$
$=\cos \left(\sin ^{3} x\right)(\cos x)-\cos 8 x^{3}(2)$
Using Leibnitz's rule
$f^{\prime}(x)=\cos \left(\sin ^{3} x\right) \frac{d}{d x}(\sin x)$
$-\cos (2 x)^{3} \frac{d}{d x}(2 x)$
$=\cos \left(\sin ^{3} x\right)(\cos x)-\cos 8 x^{3}(2)$
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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.
