What is the integral of 1/(3x+1)-1/(sqrt(4-x)) ?

The integral of 1/(3x+1)-1/(sqrt(4-x)) is 1/3 ln|3x+1|+2sqrt(4-x)+C

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integral of 1/(3x+1)-1/(sqrt(4-x))

\int \frac{1}{3 x + 1} - \frac{1}{4 - x} d x

\frac{1}{3} ln ∣ 3 x + 1 ∣ + 2 4 - x + C

Solution steps

\int \frac{1}{3 x + 1} - \frac{1}{4 - 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{1}{3 x + 1} d x - \int \frac{1}{4 - x} d x

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

\int \frac{1}{4 - x} d x = - 2 4 - x

= \frac{1}{3} ln ∣ 3 x + 1 ∣ - (- 2 4 - x)

Simplify

= \frac{1}{3} ln ∣ 3 x + 1 ∣ + 2 4 - x

Add a constant to the solution

= \frac{1}{3} ln ∣ 3 x + 1 ∣ + 2 4 - x + C

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What is the integral of 1/(3x+1)-1/(sqrt(4-x)) ?
The integral of 1/(3x+1)-1/(sqrt(4-x)) is 1/3 ln|3x+1|+2sqrt(4-x)+C