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

The integral of (3x^2+1)/(1+(x^3+x)^2) is arctan(x^3+x)+C

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

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

arctan (x^{3} + x) + C

Solution steps

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

Apply u-substitution

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

Use the common integral:

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

= arctan (u)

Substitute back

u = x^{3} + x

= arctan (x^{3} + x)

Add a constant to the solution

= arctan (x^{3} + x) + C

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