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

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

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

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

x^{2} + 4 + 3 ln (\frac{1}{2} x + 4 + x^{2}) + C

Solution steps

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

Expand

\frac{x + 3}{x ^{2} + 4} : \frac{x}{x ^{2} + 4} + \frac{3}{x ^{2} + 4}

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

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

\int \frac{3}{x ^{2} + 4} d x = 3 ln (\frac{1}{2} x + 4 + x^{2})

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

Add a constant to the solution

= x^{2} + 4 + 3 ln (\frac{1}{2} x + 4 + x^{2}) + C

\int \frac{1}{3 sin ( x ) + 4 cos ( x )} d x

\frac{d}{d x} ((5 x^{3} + 4)^{2})

\int \frac{arctan ( x )}{1 + x ^{2}} d x

7 t \cdot \frac{d y}{d t} + 3 y = t

\int \frac{2 e ^{2 x}}{e ^{2 x} + 14 e ^{x} + 48} d x

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