"Find or evaluate the integral. ∫(x3+3)ln⁡xdx" was answered by students like you

Dillard

Answered question

2021-10-21

Find or evaluate the integral.

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

Laith Petty

Skilled2021-10-22Added 103 answers

Step 1
The integral function is given as:

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

Apply the integration by parts,

\int u \cdot v d x = u \int v d x - \int u^{'} \int v d x

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

= \ln x [\frac{x^{4}}{4} + 3 x] - \int \frac{1}{x} [\frac{x^{4}}{4} + 3 x] d x

= \ln x [\frac{x^{4}}{4} + 3 x] - - \int \frac{x^{3}}{4} + 3 d x

= \ln x [\frac{x^{4}}{4} + 3 x] - [\frac{x^{4}}{16} + 3 x] + C

Step 2
The solution of the integral function is

\ln x [\frac{x^{4}}{4} + 3 x] - [\frac{x^{4}}{16} + 3 x] + C

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