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

The integral of x^3ln(x)+cos(3x+2) is 1/4 x^4ln(x)-(x^4)/(16)+1/3 sin(3x+2)+C

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

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

\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} + \frac{1}{3} sin (3 x + 2) + C

Solution steps

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

\int x^{3} ln (x) d x = \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}

\int cos (3 x + 2) d x = \frac{1}{3} sin (3 x + 2)

= \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} + \frac{1}{3} sin (3 x + 2)

Add a constant to the solution

= \frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16} + \frac{1}{3} sin (3 x + 2) + C

What is the integral of x^3ln(x)+cos(3x+2) ?
The integral of x^3ln(x)+cos(3x+2) is 1/4 x^4ln(x)-(x^4)/(16)+1/3 sin(3x+2)+C