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

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

Popular >

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{3 u} d u

= \int \frac{1}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = x^{3} + 3 x + 5

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

Add a constant to the solution

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

\int (\frac{1}{49 x ^{2} + 1}) d x

\int (x^{4} - 6 x^{3} + \frac{7}{x ^{2}}) d x

\int 2 p ln (p) d p

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

\int (x^{5} + x^{2})^{15} (5 x^{4} + 2 x) d x

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