What is the integral of 2xsqrt(4x^2-4) ?

The integral of 2xsqrt(4x^2-4) is 4/3 (x^2-1)^{3/2}+C

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integral of 2xsqrt(4x^2-4)

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

\frac{4}{3} (x^{2} - 1)^{\frac{3}{2}} + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int x 4 x^{2} - 4 d x

4 x^{2} - 4 = 2 x^{2} - 1

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

Take the constant out:

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

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

Apply u-substitution

: \int u^{2} d u

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

Apply the Power Rule

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

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

Substitute back

u = x^{2} - 1

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

Simplify

2 \cdot 2 \cdot \frac{( x ^{2} - 1 ) ^{3}}{3} : \frac{4}{3} (x^{2} - 1)^{\frac{3}{2}}

= \frac{4}{3} (x^{2} - 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{3} (x^{2} - 1)^{\frac{3}{2}} + C

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What is the integral of 2xsqrt(4x^2-4) ?
The integral of 2xsqrt(4x^2-4) is 4/3 (x^2-1)^{3/2}+C