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  • Need solution for RD Sharma maths Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 38 textbook solution.

Need solution for  RD Sharma maths Class 12 Chapter 19 Definite Integrals Exercise 19.4 (b) Question 38 textbook solution.

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Answer:- 0

Hints:-  You must know the integral rules of trignometric functions and its limits.

Given:-  \int_{0}^{2 \pi} \sin ^{100} x \cos ^{|10|} x \cdot d x

Solution : \int_{0}^{2 \pi} \sin ^{100} x \cos ^{|10|} x \cdot d x

Suppose :  f(x)=\sin ^{100} x \cos ^{\left |1 0 \right |} x

Now f(2 \pi-x)=\sin ^{100}(2 \pi-x) \cos ^{\left |1 0 \right |}(2 \pi-x)

        \begin{aligned} &=(-\sin x)^{100}(\cos x)^{|10|} \\ &=\sin ^{100} x \cdot \cos ^{|10|} x \\ &=f(x) \end{aligned}

\begin{aligned} \therefore I &=\int_{0}^{2 \pi} \sin ^{100} x \cos ^{10} x \cdot d x \\ &=2 \int_{0}^{\pi} \sin ^{100} x \cos ^ {10} x \cdot d x \end{aligned}

Again

        \begin{aligned} f(\pi-x) &=\sin ^{100}(\pi-x) \cdot \cos ^{|10|}(\pi-x) \\ &=\sin ^{100} x \cdot(-\cos x)^{|10|}x \\ &=-\sin ^{100} x \cdot \cos ^{10}x \\ &=-f(x) \end{aligned}

Its and odd function

           \begin{aligned} \therefore I &=2 \times 0 \\ &=0 \end{aligned}

     

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