What is the integral of ((-3)/(1-2x)+2sin(5-6x)) ?

The integral of ((-3)/(1-2x)+2sin(5-6x)) is 3/2 ln|1-2x|+1/3 cos(5-6x)+C

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integral of ((-3)/(1-2x)+2sin(5-6x))

\int (\frac{- 3}{1 - 2 x} + 2 sin (5 - 6 x)) d x

\frac{3}{2} ln ∣ 1 - 2 x ∣ + \frac{1}{3} cos (5 - 6 x) + C

Solution steps

\int \frac{- 3}{1 - 2 x} + 2 sin (5 - 6 x) d x

Simplify

= \int - \frac{3}{1 - 2 x} + 2 sin (5 - 6 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 \frac{3}{1 - 2 x} d x + \int 2 sin (5 - 6 x) d x

\int \frac{3}{1 - 2 x} d x = - \frac{3}{2} ln ∣ 1 - 2 x ∣

\int 2 sin (5 - 6 x) d x = \frac{1}{3} cos (5 - 6 x)

= - (- \frac{3}{2} ln ∣ 1 - 2 x ∣) + \frac{1}{3} cos (5 - 6 x)

Simplify

= \frac{3}{2} ln ∣ 1 - 2 x ∣ + \frac{1}{3} cos (5 - 6 x)

Add a constant to the solution

= \frac{3}{2} ln ∣ 1 - 2 x ∣ + \frac{1}{3} cos (5 - 6 x) + C

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What is the integral of ((-3)/(1-2x)+2sin(5-6x)) ?
The integral of ((-3)/(1-2x)+2sin(5-6x)) is 3/2 ln|1-2x|+1/3 cos(5-6x)+C