Solve the indefinite integral of y=x^3/(x^4+1)

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To calculate the indefinite integral, we'll use the substitution method.

We'll note y = f(x)

We'll calculate Integral of f(x) = y = x^3/(x^4+1).

We notice that if we'll differentiate x^4+1, we'll get 4x^3.

So, we'll note x^4+1 = t

(x^4+1)'dx = dt

(4x^3)dx = dt => (x^3)dx = dt/4

We'll re-write the integral in the variable t:

Int  (x^3)dx/(x^4+1) = Int dt / 4t

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

(1/4)*Int dt / t = (1/4)*ln t + C

But x^4+1 = t.

Int  (x^3)dx/(x^4+1) = (1/4)*ln(x^4+1) + C, where C is a family of constants.

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