What is the integral of ln(e^4x^3) ?

The integral of ln(e^4x^3) is 3xln(x)+x+C

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integral of ln(e^4x^3)

\int ln (e^{4} x^{3}) d x

3 x ln (x) + x + C

Solution steps

\int ln (e^{4} x^{3}) d x

Simplify

ln (e^{4} x^{3}) : ln (x^{3}) + 4

= \int ln (x^{3}) + 4 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int ln (x^{3}) d x + \int 4 d x

\int ln (x^{3}) d x = 3 x ln (x) - 3 x

\int 4 d x = 4 x

= 3 x ln (x) - 3 x + 4 x

Simplify

= 3 x ln (x) + x

Add a constant to the solution

= 3 x ln (x) + x + C

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