What is the integral of (a+4)/(4a+7) ?

The integral of (a+4)/(4a+7) is 1/4 a+9/16 ln|4a+7|+C

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integral of (a+4)/(4a+7)

\int \frac{a + 4}{4 a + 7} d a

\frac{1}{4} a + \frac{9}{16} ln ∣ 4 a + 7 ∣ + C

Solution steps

\int \frac{a + 4}{4 a + 7} d a

Expand

\frac{a + 4}{4 a + 7} : \frac{a}{4 a + 7} + \frac{4}{4 a + 7}

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{a}{4 a + 7} d a + \int \frac{4}{4 a + 7} d a

\int \frac{a}{4 a + 7} d a = \frac{1}{4} a - \frac{7}{16} ln ∣ 4 a + 7 ∣

\int \frac{4}{4 a + 7} d a = ln ∣ 4 a + 7 ∣

= \frac{1}{4} a - \frac{7}{16} ln ∣ 4 a + 7 ∣ + ln ∣ 4 a + 7 ∣

Add similar elements:

- \frac{7}{16} ln ∣ 4 a + 7 ∣ + ln ∣ 4 a + 7 ∣ = \frac{9}{16} ln ∣ 4 a + 7 ∣

= \frac{1}{4} a + \frac{9}{16} ln ∣ 4 a + 7 ∣

Add a constant to the solution

= \frac{1}{4} a + \frac{9}{16} ln ∣ 4 a + 7 ∣ + C

What is the integral of (a+4)/(4a+7) ?
The integral of (a+4)/(4a+7) is 1/4 a+9/16 ln|4a+7|+C