integral of (10)/((25x^2+1)^2) — step-by-step solution

Let's integrate:

\int(\frac{10}{{(25 x^{2} + 1)}^{2}})dx

The denominator

25 x^{2} + 1

is strictly positive on ℝ. Let

u = 5 x

, so

du = 5 dx

The quotient rule and the derivative of atan give

{(\frac{u}{1 + u^{2}} + atan (u))}^{'}

2 \cdot \frac{1}{{(1 + u^{2})}^{2}}

: an antiderivative of

\frac{1}{{(1 + u^{2})}^{2}}

\frac{\frac{u}{1 + u^{2}} + atan (u)}{2}

Apply the factor

1

, then substitute

u

back:

\frac{5 x}{1 + {(5 x)}^{2}} + atan (5 x)

\frac{5 x}{1 + 5^{2} \cdot x^{2}} + atan (5 x)

\frac{5 x}{1 + 25 x^{2}} + atan (5 x)

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

\frac{5 x}{25 x^{2} + 1} + atan (5 x) + C