AP EAMCET · Maths · Definite Integration
Choose the correct option regarding the following definite integrals
(i) \(\int_0^{\pi / 2} \sin ^m(x) \cos (x) d x=\frac{1}{m+1}\)
(ii) \(\int_0^{\pi / 2} \sin (x) \cos ^n(x) d x=\frac{1}{n+1}\)
- A (i) is true, (ii) is false
- B (i) is false, (ii) is true
- C Both (i) and (ii) are false
- D Both (i) and (ii) are true
Answer & Solution
Correct Answer
(D) Both (i) and (ii) are true
Step-by-step Solution
Detailed explanation
(i) \(\int_0^{\pi / 2} \sin ^m x \cdot \cos x d x\) Put, \(\sin x=t\) \[ \begin{aligned} \cos x d x & =d t=\int_0^{\pi / 2} t^m d t \\ & =\left[\frac{t^{m+1}}{m+1}\right]_0^{\pi / 2}=\left[\frac{(\sin x)^{m+1}}{m+1}\right]_0^{\pi / 2} \end{aligned} \]…
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