What is the integral of (6x^2+1)ln(5x^3) ?

The integral of (6x^2+1)ln(5x^3) is 2ln(5)x^3+ln(5)x+6x^3ln(x)+3xln(x)-2x^3-3x+C

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integral of (6x^2+1)ln(5x^3)

\int (6 x^{2} + 1) ln (5 x^{3}) d x

2 ln (5) x^{3} + ln (5) x + 6 x^{3} ln (x) + 3 x ln (x) - 2 x^{3} - 3 x + C

Solution steps

\int (6 x^{2} + 1) ln (5 x^{3}) d x

Apply Integration By Parts

: ln (5 x^{3}) (2 x^{3} + x) - \int 3 (2 x^{2} + 1) d x

= ln (5 x^{3}) (2 x^{3} + x) - \int 3 (2 x^{2} + 1) d x

\int 3 (2 x^{2} + 1) d x = 2 x^{3} + 3 x

= ln (5 x^{3}) (2 x^{3} + x) - (2 x^{3} + 3 x)

Simplify

ln (5 x^{3}) (2 x^{3} + x) - (2 x^{3} + 3 x) : 2 ln (5) x^{3} + ln (5) x + 6 x^{3} ln (x) + 3 x ln (x) - 2 x^{3} - 3 x

= 2 ln (5) x^{3} + ln (5) x + 6 x^{3} ln (x) + 3 x ln (x) - 2 x^{3} - 3 x

Add a constant to the solution

= 2 ln (5) x^{3} + ln (5) x + 6 x^{3} ln (x) + 3 x ln (x) - 2 x^{3} - 3 x + C

\int (sec^{2} (π x) + e^{- 3 x}) d x

\int \frac{sin ( 2 π ) x}{3} d x

\int - \frac{e ^{x}}{x} d x

\int (e^{x}) \cdot sin (n x) d x

What is the integral of (6x^2+1)ln(5x^3) ?
The integral of (6x^2+1)ln(5x^3) is 2ln(5)x^3+ln(5)x+6x^3ln(x)+3xln(x)-2x^3-3x+C