What is the integral of (3x+1)/(sqrt(5x^2+1)) ?

The integral of (3x+1)/(sqrt(5x^2+1)) is 3/5 sqrt(5x^2+1)+1/(sqrt(5))ln|sqrt(5)x+sqrt(1+5x^2)|+C

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integral of (3x+1)/(sqrt(5x^2+1))

\int \frac{3 x + 1}{5 x ^{2} + 1} d x

\frac{3}{5} 5 x^{2} + 1 + \frac{1}{5} ln 5 x + 1 + 5 x^{2} + C

Solution steps

\int \frac{3 x + 1}{5 x ^{2} + 1} d x

Expand

\frac{3 x + 1}{5 x ^{2} + 1} : \frac{3 x}{5 x ^{2} + 1} + \frac{1}{5 x ^{2} + 1}

= \int \frac{3 x}{5 x ^{2} + 1} + \frac{1}{5 x ^{2} + 1} 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 x}{5 x ^{2} + 1} d x + \int \frac{1}{5 x ^{2} + 1} d x

\int \frac{3 x}{5 x ^{2} + 1} d x = \frac{3}{5} 5 x^{2} + 1

\int \frac{1}{5 x ^{2} + 1} d x = \frac{1}{5} ln 5 x + 1 + 5 x^{2}

= \frac{3}{5} 5 x^{2} + 1 + \frac{1}{5} ln 5 x + 1 + 5 x^{2}

Add a constant to the solution

= \frac{3}{5} 5 x^{2} + 1 + \frac{1}{5} ln 5 x + 1 + 5 x^{2} + C

x \to - 1 lim (- 2)

f (x) = 2 x^{2} + 3 x - 9

\int \frac{ln ( x ^{8} )}{x} d x

\frac{1}{3 x - 1}

\frac{d y}{d x} + y = x, y (0) = 4

What is the integral of (3x+1)/(sqrt(5x^2+1)) ?
The integral of (3x+1)/(sqrt(5x^2+1)) is 3/5 sqrt(5x^2+1)+1/(sqrt(5))ln|sqrt(5)x+sqrt(1+5x^2)|+C