Question:
If $I_{1} = \int\limits^{3\pi}_{0}f\left(\cos^{2}\,x\right)dx$ and $I_{2} = \int\limits^{\pi}_{0} f\left(\cos^{2}\,x\right)dx$, then
If $I_{1} = \int\limits^{3\pi}_{0}f\left(\cos^{2}\,x\right)dx$ and $I_{2} = \int\limits^{\pi}_{0} f\left(\cos^{2}\,x\right)dx$, then
Updated On: Jun 18, 2022
- $I_1 = I_2$
- $3I_1 = I_2$
- $I_1 = 3I_2$
- $I_1 = 5I_2$
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The Correct Option is C
Solution and Explanation
$I_{1}=3 \int\limits_{0}^{\pi} f\left(\cos ^{2} x\right) d x=3 I_{2}$
$[\because$ period is $\pi]$
$[\because$ period is $\pi]$
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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.
