What is the indefinite integral of f(x)=x^2(x^3+1)^4 ?

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To evaluate the indefinite integral of the given function, we'll apply the substitution technique.

We'll note x^3 + 1 = t.

We'll differentiate both sides:

3x^2dx = dt

x^2dx = dt/3

We'll re-write the integral with the new variable t:

Int t^4*(dt/3) = (1/3)*Int t^4dt

(1/3)Int t^4dt = t^5/5 + C

But t = x^3 + 1

**Int x^2(x^3+1)^4 = (x^3 + 1)^5/5 + C**

**The final result is (x^3 + 1)^5/15 + C**

To find the indefinite integral of f(x)=x^2(x^3+1)^4, let us first denote x^3 + 1 = t

=> dt/dx = 3x^2

=> x^2 dx = (1/3) dt

Int [ x^2(x^3+1)^4 dx]

=> Int [ (1/3) t^4 dt]

=> t^5 / (3*5)

=> t^5 / 15 + C

replace t with x^3 + 1

=> [(x^3 + 1)^5]/ 15 + C

**The required integral is [(x^3 + 1)^5]/ 15 + C**

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