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Need solution for RD Sharma Maths Class 12 Chapter 19 Definite Integrals Excercise 19.3 Question 23

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Answer:  0

Hint: We will use the property of definite integrals.

Given:  \int_{0}^{\pi} \cos x|\cos x| d x


                I=\int_{0}^{\pi} \cos x|\cos x| d x                                                       … (i)

Consider,  f(x)=\cos x|\cos x|

Now, use the property:   \int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x

                 I=\int_{0}^{\pi} \cos (\pi-x)|\cos (\pi-x)| d x \\

                =\int_{0}^{\pi}-\cos (x)|\cos (x)| d x \\

                \begin{aligned} & &I=\int_{0}^{\pi}-\cos (x)|\cos (x)| d x \end{aligned}                                       … (ii)

Adding (i) and (ii) , we get

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


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