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

The integral of (2x)/(sqrt(1+x^2)) is 2sqrt(1+x^2)+C

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integral of (2x)/(sqrt(1+x^2))

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

2 1 + x^{2} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{2 u} d u

= 2 \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 2 \cdot \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + x^{2}

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

Simplify

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

= 2 1 + x^{2}

Add a constant to the solution

= 2 1 + x^{2} + C

(2, 5), (8, 12)

\int e^{2 x + y} d x

\int \frac{4 - 9 x ^{2}}{x} d x

x \to 0 - lim (\frac{x ^{2}}{2} - \frac{8}{x})

x \to 2 lim (∣ x - 2 ∣)

What is the integral of (2x)/(sqrt(1+x^2)) ?
The integral of (2x)/(sqrt(1+x^2)) is 2sqrt(1+x^2)+C