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
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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