What is the integral of-2ln(3/2 x)-cos(3x) ?

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

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integral of-2ln(3/2 x)-cos(3x)

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

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

Solution steps

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

Apply Integration By Parts

: x (- 2 ln (\frac{3}{2} x) - cos (3 x)) - \int - 2 + 3 x sin (3 x) d x

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

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

= x (- 2 ln (\frac{3}{2} x) - cos (3 x)) - (- 2 x - x cos (3 x) + \frac{1}{3} sin (3 x))

Simplify

x (- 2 ln (\frac{3}{2} x) - cos (3 x)) - (- 2 x - x cos (3 x) + \frac{1}{3} sin (3 x)) : - 2 x ln (\frac{3}{2} x) + 2 x - \frac{1}{3} sin (3 x)

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

Add a constant to the solution

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

\int \frac{x e ^{x}}{1 + e ^{x}} d x

\int 7 e^{- t} d t

\frac{d}{d x} (\frac{1 - 2 e ^{x}}{1 - 2 cos ( x )})

(x + 1) y^{^{'}} = x^{2} - y

\int \frac{9}{x ^{2} + 1} d x

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