Integral.Evaluate the indefinite integral of y=(x^3+5)*ln x?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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This problem requires integration by parts:

We'll put u = ln x.

We'll differentiate both sides and we'll get:

du = dx/x

We'll put dv = (x^3+5)dx

v = Int (x^3+5)dx

Int (x^3+5)dx = Int x^3dx + 5Int dx

Int (x^3+5)dx = x^4/4 + 5x + C

v = x^4/4 + 5x

We'll write the formula of integration by parts:

Int u dv = u*v - Int v du

Int (x^3+5)*ln x dx = (ln x)*(x^4/4 + 5x) - Int (x^4/4 + 5x)dx/x

Int (x^4/4 + 5x)dx/x = Int x^4dx/4x + Int 5xdx/x

We'll simplify and we'll get:

Int (x^4/4 + 5x)dx/x = (1/4)Int x^3dx + 5Int dx

Int x^4dx/4x + Int 5xdx/x = (1/4)*(x^4/4) + 5x

Int (x^3+5)*ln x dx = (ln x)*(x^4/4 + 5x) - x^4/16 - 5x + C

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