integral of (3x)/(1+x^2) — step-by-step solution

Let's compute the integral:

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

For the math expression:

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

, since (ln(a))' = a'/a, we try differentiating

\ln (1 + x^{2})

Let's perform the differentiation:

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

Apply the differentiation rule:

\frac{d}{d x} (\ln (f)) = \frac{d}{d x} (f) \cdot \frac{1}{f}

with f =

1 + x^{2}

ⓘ The derivative of ln(x) is 1/x:

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

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

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

ⓘ Multiplying a fraction by a term multiplies its numerator:

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

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

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

■The derivative of a sum is the sum of the derivatives:

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

0 + \frac{d}{d x} (x^{2})

\frac{0 + \frac{d}{d x} (x^{2})}{1 + x^{2}}

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (x^{2})

\frac{d}{d x} (x^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

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

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

ⓘ The derivative of

\frac{3}{2} \cdot \ln (1 + x^{2})

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

, so

\frac{3}{2} \cdot \ln (1 + x^{2})

is an antiderivative of

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

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

\frac{3}{2} \cdot \ln (1 + x^{2})

\frac{3}{2} \cdot \ln (1 + x^{2})

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{3}{2} \cdot \ln (1 + x^{2})

\frac{3 \cdot \ln (1 + x^{2})}{2}

\frac{3 \cdot \ln (1 + x^{2})}{2}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

\frac{3 \cdot \ln (x^{2} + 1)}{2} + C