Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
The value of $\int_0^{2 x} \cos ^{99} x d x$ is
A
1
B
-1
C
99
✓
$0$
✓
Answer
Correct option: D.
$0$
(D) $\int_0^{2 \pi} \cos ^{99} x d x=2 \int_0^\pi \cos ^{99} x d x$ $\ldots\left[\because \int_0^{2 a } f (x) d x=2 \int_0^{ a } f (x) d x\right.$, if $\left.f (2 a -x)= f (x)\right]$ Let $I _1=\int_0^\pi \cos ^{99} x d x$ $\Rightarrow I _1=-\int_0^\pi \cos ^{99} x d x \ldots .\left[\because \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$ $\Rightarrow I _1=- I _1 \Rightarrow 2 I _1=0 \Rightarrow I _1=0$ $\therefore \quad \int_0^{2 \pi} \cos ^{99} x d x=2(0)=0$
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