If $ f(x)=\int_{1}^{x}{\sqrt{4-{{t}^{2}}}}\,\,\,dt, $ then real roots of the equation $ x-f'(x)=0 $ are
- $ \pm \,\,1 $
\(\pm \,\,\sqrt{2}\)
- $ 0 $ and $ 1 $
- $ \pm \,\,2 $
The Correct Option is B
Solution and Explanation
The correct option is(B): ±√2.
Given, \(f(x)=\int_{1}^{x}{\sqrt{4-{{t}^{2}}}}\,\,dt\)
On differentiating w. r. t. x, we get
\(f'(x)=\sqrt{4-{{x}^{2}}}\) (1)
\(\therefore\) \(x-f'(x)=x-\sqrt{4-{{x}^{2}}}=0\)
\(\Rightarrow\) \(x=\sqrt{4-{{x}^{2}}}\)
\(\Rightarrow\) \({{x}^{2}}=4-{{x}^{2}}\)
\(\Rightarrow\) \(2{{x}^{2}}=4\)
\(\Rightarrow\) \({{x}^{2}}=2\)
\(\Rightarrow\) \(x=\pm 2\)
Hence, real roots of
\(\{x-f'(x)\}\) and \(\pm \sqrt{2}\) .
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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.
