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Provide Solution For R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.25 Question 17 Maths Textbook Solution.

Answers (1)

Answer: e^{x^{2}}\left(x^{2}-1\right)+c

Hint:Take, x^{2}=t

        So we get 2 x d x=d t

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

                    =\int 2 x x^{2} e^{x^{2}} d x=\int t . e^{t} d t

By integrating w.r.t ‘t’ taking the first function as t and second function as e^{t}

                  \begin{aligned} &=\int t . e^{t} d t=t \int e^{t} d t-\int\left(\frac{d}{d t} t \int e^{t} d t\right) d t \\ &=t e^{t}-e^{t}+c \end{aligned}

Now by replacing t with x^{2}

                  x^{2} e^{x^{2}}-e^{x^{2}}+c

By taking e^{x^{2}} as common


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