Evaluate using substitution of \int _{ }^{ }\frac{2x}{(x^2+1)}

\ln{\left(x^{2}+1\right)}+C,C \in \mathbb{R}

Solve int _ ^ frac2x(x^2+1) - Socratic AI (ScanMath)

Evaluate

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

Remove the parentheses

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

Rewrite the expression

\int 2 \times \frac{x}{x ^{2} + 1} d x

Use the property of integral \int k f (x) d x = k \int f (x) d x

2 \times \int \frac{x}{x ^{2} + 1} d x

Use the substitution d x = \frac{1}{2 x} d t to transform the integral

More Steps

2 \times \int \frac{x}{x ^{2} + 1} \times \frac{1}{2 x} d t

Simplify

More Steps

2 \times \int \frac{1}{2 x ^{2} + 2} d t

Use the substitution t = x^{2} to transform the integral

2 \times \int \frac{1}{2 t + 2} d t

Rewrite the expression

2 \times \int \frac{1}{2} \times \frac{1}{t + 1} d t

Use the property of integral \int k f (x) d x = k \int f (x) d x

2 \times \frac{1}{2} \times \int \frac{1}{t + 1} d t

Multiply the numbers

More Steps

1 \times \int \frac{1}{t + 1} d t

Simplify

\int \frac{1}{t + 1} d t

Use the property of integral \int \frac{1}{a x + b} d x = \frac{1}{a} ln ∣ a x + b ∣

ln (∣ t + 1 ∣)

Substitute back

ln (x^{2} + 1)

When the expression in absolute value bars is not negative, remove the bars

ln (x^{2} + 1)

Solution

ln (x^{2} + 1) + C, C \in R