What is the integral of (4x+1)/(4x^2+2x+3) ?

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

Popular >

integral of (4x+1)/(4x^2+2x+3)

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = 4 x^{2} + 2 x + 3

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

Add a constant to the solution

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

2 t \cdot \frac{d y}{d t} + 5 y = t

y = \frac{8 x}{e ^{x}}

x \to 1 lim (e^{5 x^{3} - 5 x})

\int_{0}^{6} \frac{2 x}{( x - 7 ) ^{2}} d x

\frac{r}{r ^{2} + 5}

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