What is the integral of (3x+4)/((x^2+4)(3-x)) ?

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

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integral of (3x+4)/((x^2+4)(3-x))

\int \frac{3 x + 4}{( x ^{2} + 4 ) ( 3 - x )} d x

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

Solution steps

\int \frac{3 x + 4}{( x ^{2} + 4 ) ( 3 - x )} d x

Take the partial fraction of

\frac{3 x + 4}{( x ^{2} + 4 ) ( 3 - x )} : \frac{x}{x ^{2} + 4} - \frac{1}{x - 3}

= \int \frac{x}{x ^{2} + 4} - \frac{1}{x - 3} 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{x}{x ^{2} + 4} d x - \int \frac{1}{x - 3} d x

\int \frac{x}{x ^{2} + 4} d x = \frac{1}{2} ln x^{2} + 4

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

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

Add a constant to the solution

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

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