What is the integral of ln(x)(3x^2+7) ?

The integral of ln(x)(3x^2+7) is x^3ln(x)-(x^3)/3+7(xln(x)-x)+C

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integral of ln(x)(3x^2+7)

\int ln (x) (3 x^{2} + 7) d x

x^{3} ln (x) - \frac{x ^{3}}{3} + 7 (x ln (x) - x) + C

Solution steps

\int ln (x) (3 x^{2} + 7) d x

Expand

ln (x) (3 x^{2} + 7) : 3 x^{2} ln (x) + 7 ln (x)

= \int 3 x^{2} ln (x) + 7 ln (x) 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 3 x^{2} ln (x) d x + \int 7 ln (x) d x

\int 3 x^{2} ln (x) d x = x^{3} ln (x) - \frac{x ^{3}}{3}

\int 7 ln (x) d x = 7 (x ln (x) - x)

= x^{3} ln (x) - \frac{x ^{3}}{3} + 7 (x ln (x) - x)

Add a constant to the solution

= x^{3} ln (x) - \frac{x ^{3}}{3} + 7 (x ln (x) - x) + C

\int x^{2} (2 x - 1) d x

\int \frac{e ^{9 x}}{e ^{9 x} + 7} d x

\int 5 e^{x} + \frac{4}{x} d x

\int (e^{9 x}) d x

\int x^{\frac{3}{4}} - \frac{1}{x ^{\frac{3}{4}}} d x

What is the integral of ln(x)(3x^2+7) ?
The integral of ln(x)(3x^2+7) is x^3ln(x)-(x^3)/3+7(xln(x)-x)+C