MHT CET · Maths · Definite Integration
\(\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x d x=\)
- A \(\frac{\pi}{4}\)
- B \(\frac{\pi}{3}\)
- C \(\frac{\pi}{2}\)
- D \(\frac{3 \pi}{4}\)
Answer & Solution
Correct Answer
(C) \(\frac{\pi}{2}\)
Step-by-step Solution
Detailed explanation
Let \(f(x)=\sin ^{2} x\)
\(\therefore \mathrm{f}(-\mathrm{x})=[\sin (-\mathrm{x})]^{2}=\sin ^{2} \mathrm{x}\)
Thus \(\sin ^{2} x\) is an even function.
\(\therefore \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x d x=2 \int_{0}^{\frac{\pi}{2}} \sin ^{2} x d x\)
\(=2 \int_{0}^{\frac{\pi}{2}} \frac{1}{2}(1-\cos 2 x) d x=\left[x-\frac{\sin 2 x}{2}\right]_{0}^{\frac{\pi}{2}}\)
\(=\left(\frac{\pi}{2}-0\right)-\left(\frac{\sin \pi}{2}-\frac{\sin 0}{2}\right)=\frac{\pi}{2}\)
\(\therefore \mathrm{f}(-\mathrm{x})=[\sin (-\mathrm{x})]^{2}=\sin ^{2} \mathrm{x}\)
Thus \(\sin ^{2} x\) is an even function.
\(\therefore \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x d x=2 \int_{0}^{\frac{\pi}{2}} \sin ^{2} x d x\)
\(=2 \int_{0}^{\frac{\pi}{2}} \frac{1}{2}(1-\cos 2 x) d x=\left[x-\frac{\sin 2 x}{2}\right]_{0}^{\frac{\pi}{2}}\)
\(=\left(\frac{\pi}{2}-0\right)-\left(\frac{\sin \pi}{2}-\frac{\sin 0}{2}\right)=\frac{\pi}{2}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(\mathrm{y}=\tan ^{-1}\left(\frac{12 x-64 x^3}{1-48 x^2}\right)\), then \(\frac{\mathrm{dy}}{\mathrm{d} x}=\)MHT CET 2025 Medium
- On which of the following lines lies the point of intersection of the line, \(\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}\) and the plane \(x+y+z=2\) ?MHT CET 2024 Easy
- A linguistic club of a certain Institute consists of 6 girls and 4 boys. A team of 4 members to be selected from this group including the selection of a Captain (from among these 4 members) for the team. If the team has to include atmost one boy, the number of ways of selecting the team isMHT CET 2023 Easy
- If A, B, C and D are and respectively, then the volume of the parallelepiped with AB, AC and AD as the co-terminus edges, is ….cubic units.MHT CET 2019 Medium
- If \(P\) is a point on the segment \(A B\) of length \(12 \mathrm{~cm}\), then the position of \(P\) for \(A P^{2}+B P^{2}\) to be minimum is such thatMHT CET 2020 Medium
- If \(\sin \left(\frac{x+y}{x-y}\right)=\tan \frac{\pi}{5}\), then \(\frac{d y}{d x}=\)MHT CET 2020 Medium
More PYQs from MHT CET
- Which of the following aldehydes is less reactive towards nucleophilic addition reaction?MHT CET 2021 Easy
- When a d.c voltage of \(200 \mathrm{~V}\) is applied to a coil of self-inductance \(\left(\frac{2 \sqrt{3}}{\pi}\right) \mathrm{H}\), a current of 1A flows through it. But by replacing d.c. source with a.c. source of \(200 \mathrm{~V}\), the current in the coil is reduced to \(0.5 \mathrm{~A}\). Then the frequency of a.c. supply isMHT CET 2021 Hard
- A signal of frequency \(3 \mathrm{kHz}\) is amplitude modulated on a carrier wave of frequency \(2 \cdot 5 \mathrm{MHz}\). The upper and lower sideband frequencies in the resultant signal areMHT CET 2020 Easy
- The susceptibility of a magnetic material is positive and small. The material isMHT CET 2020 Easy
- In reverse bias \(p n\) -junction diode depletion layer widthMHT CET 2011 Easy
- The area (in sq. units) bounded by the curves \(y=\sqrt{x}, 2 y-x+3=0\), \(\mathrm{X}\)-axis and lying in the first quadrant, isMHT CET 2022 Easy