What is the integral of ((10x)/(1+5x^2)) ?

The integral of ((10x)/(1+5x^2)) is ln|1+5x^2|+C

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integral of ((10x)/(1+5x^2))

\int (\frac{10 x}{1 + 5 x ^{2}}) d x

ln 1 + 5 x^{2} + C

Solution steps

\int \frac{10 x}{1 + 5 x ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int \frac{x}{1 + 5 x ^{2}} d x

Apply u-substitution

: \int \frac{1}{10 u} d u

= 10 \cdot \int \frac{1}{10 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \frac{1}{10} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 10 \cdot \frac{1}{10} ln ∣ u ∣

Substitute back

u = 1 + 5 x^{2}

= 10 \cdot \frac{1}{10} ln 1 + 5 x^{2}

Simplify

10 \cdot \frac{1}{10} ln 1 + 5 x^{2} : ln 1 + 5 x^{2}

= ln 1 + 5 x^{2}

Add a constant to the solution

= ln 1 + 5 x^{2} + C

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

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\frac{d}{d x} (\frac{e ^{x}}{3 x ^{2}})

x \to 0 lim (6 x^{2} (cot (3 x)) (csc (2 x)))

\int_{0}^{1} (x - x^{2})^{2} d x

What is the integral of ((10x)/(1+5x^2)) ?
The integral of ((10x)/(1+5x^2)) is ln|1+5x^2|+C