integral of x/((x^2+7)^2) — step-by-step solution

1) Let's integrate:

\int \frac{x}{{(x^{2} + 7)}^{2}} d x

For the math expression:

\frac{x}{{(x^{2} + 7)}^{2}}

, since (a^n)' = n*a'*a^(n - 1), we try differentiating

{(x^{2} + 7)}^{- 2 + 1}

2) Let's perform the differentiation:

\frac{d}{d x} ({(x^{2} + 7)}^{- 2 + 1})

ⓘ The exponent is computed before the power:

- 2 + 1

-1

\frac{d}{d x} ({(x^{2} + 7)}^{-1})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

x^{2} + 7

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(x^{2} + 7)}^{-1})

-1 \cdot \frac{d}{d x} (x^{2} + 7) \cdot {(x^{2} + 7)}^{-2}

-1 \cdot \frac{d}{d x} (x^{2} + 7) \cdot {(x^{2} + 7)}^{-2}

ⓘ Multiplying by -1 gives the opposite:

-1 \cdot \frac{d}{d x} (x^{2} + 7) \cdot {(x^{2} + 7)}^{-2}

- \frac{d}{d x} (x^{2} + 7) \cdot {(x^{2} + 7)}^{-2}

- \frac{d}{d x} (x^{2} + 7) \cdot {(x^{2} + 7)}^{-2}

ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

- \frac{d}{d x} (x^{2} + 7) \cdot {(x^{2} + 7)}^{-2}

- \frac{\frac{d}{d x} (x^{2} + 7)}{{(x^{2} + 7)}^{2}}

- \frac{\frac{d}{d x} (x^{2} + 7)}{{(x^{2} + 7)}^{2}}

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

\frac{d}{d x} (x^{2} + 7)

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

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

ⓘ Adding 0 changes nothing:

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

\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}{{(x^{2} + 7)}^{2}}

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

ⓘ The derivative of

- \frac{1}{2} \cdot \frac{1}{x^{2} + 7}

\frac{x}{{(x^{2} + 7)}^{2}}

, so

- \frac{1}{2} \cdot \frac{1}{x^{2} + 7}

is an antiderivative of

\frac{x}{{(x^{2} + 7)}^{2}}

\int \frac{x}{{(x^{2} + 7)}^{2}} d x

- \frac{1}{2} \cdot \frac{1}{x^{2} + 7}

- \frac{1}{2} \cdot \frac{1}{x^{2} + 7}

ⓘ To multiply fractions, multiply the numerators together and the denominators together:

- \frac{1}{2} \cdot \frac{1}{x^{2} + 7}

- \frac{1 \cdot 1}{2 \cdot (x^{2} + 7)}

- \frac{1}{2 \cdot (x^{2} + 7)}

- \frac{1}{2 \cdot (x^{2} + 7)}

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

- \frac{1}{2 \cdot (x^{2} + 7)} + C