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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 18 Maths Textbook Solution.

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Answer: \frac{1}{2} x^{2} \sin x^{2}+\frac{1}{2} \cos x^{2}+c

Hint: Take x^{2}=t

          So we get 2 x d x=d t

                       x d x=\frac{d t}{2}

Given: Let I=\int x^{3} \cos x^{2} d x

Solution: I=\int x^{3} \cos x^{2} d x

We can write it as

             \begin{aligned} &\int x x^{2} \cos x^{2} d x \\ &=\frac{1}{2} \int t \cos t d t=\frac{1}{2}\left(t \int \cos t d t-\int\left[\frac{d t}{d t} \int \cos t d t\right] d t\right) \\ &=\frac{1}{2}\left(t \sin t-\int \sin t d t\right) \end{aligned}

Now by integrating the second part

            =\frac{1}{2}(t \sin t+\cos t)+c

Substituting the value of t as x^{2}

         =\frac{1}{2} x^{2} \sin x^{2}+\frac{1}{2} \cos x^{2}+c

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