If f(x) = ∫x0 t(sin x-sin t)dt then
If f(x) = ∫x0 t(sin x-sin t)dt then
- $f '''(x)+f ''(x)=\sin x$
- $f '''(x)+f''(x)-f (x)=\cos x$
- $f'''(x)+f '(x)=\cos x-2x \sin x$
- $f '''(x)-f ''(x)=\cos x-2x \sin x$
The Correct Option is C
Solution and Explanation
\(f\left(x\right) = \int^{^x}_{_0}t\left(sin\,x - sin\,t\right)dt\)
\(f\left(x\right) = sinx \int^{^x}_{_0}t\,dt - \int^{^x}_{_0} t \,sin\,tdt\)
\(f'\left(x\right) = \left(sinx\right) x +cosx \int^{^x}_{_0} t \,dt-x\,sinx\)
\(f'\left(x\right) = cosx \int^{^x}_{_0} tdt\) = \(x\cos x\)
\(f''\left(x\right) = \left(cosx\right) x - \left(sinx\right) \int^{^x}_{_0} tdt\)
\(f'''\left(x\right) = x\left(-sinx\right) + cosx-\left(sinx\right)x-\left(cosx\right) \int^{^x}_{_0} tdt\)
\(f'''\left(x\right)+f'\left(x\right) = cosx-2x\,sinx\)
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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.
