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

The integral of sqrt(1+x^2)2x is 2/3 (1+x^2)^{3/2}+C

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int u^{2} d u

= 2 \cdot \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= 2 \cdot \frac{u ^{3}}{3}

Substitute back

u = 1 + x^{2}

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

Simplify

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

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

Add a constant to the solution

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

\frac{1}{( s + 20 ) [ ( s + 2 ) ^{2} + 1 ]}

x \to \frac{π}{2} lim (x sec^{2} (x))

n = 0 \sum \infty \frac{3 ^{n}}{1000}

\frac{d y}{d x} = x^{2} y

\frac{d}{d x} (\frac{4 x - 4 1 + x}{x ^{2} - 1})

What is the integral of sqrt(1+x^2)2x ?
The integral of sqrt(1+x^2)2x is 2/3 (1+x^2)^{3/2}+C