What is the integral of x^6ln(x)+4xsin(3x) ?

The integral of x^6ln(x)+4xsin(3x) is 1/7 x^7ln(x)-(x^7)/(49)+4(-1/3 xcos(3x)+1/9 sin(3x))+C

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integral of x^6ln(x)+4xsin(3x)

\int x^{6} ln (x) + 4 x sin (3 x) d x

\frac{1}{7} x^{7} ln (x) - \frac{x ^{7}}{49} + 4 (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x)) + C

Solution steps

\int x^{6} ln (x) + 4 x sin (3 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 x^{6} ln (x) d x + \int 4 x sin (3 x) d x

\int x^{6} ln (x) d x = \frac{1}{7} x^{7} ln (x) - \frac{x ^{7}}{49}

\int 4 x sin (3 x) d x = 4 (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x))

= \frac{1}{7} x^{7} ln (x) - \frac{x ^{7}}{49} + 4 (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x))

Add a constant to the solution

= \frac{1}{7} x^{7} ln (x) - \frac{x ^{7}}{49} + 4 (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x)) + C

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What is the integral of x^6ln(x)+4xsin(3x) ?
The integral of x^6ln(x)+4xsin(3x) is 1/7 x^7ln(x)-(x^7)/(49)+4(-1/3 xcos(3x)+1/9 sin(3x))+C