What is the integral of (2x)/(sqrt(4x^2+7)) ?

The integral of (2x)/(sqrt(4x^2+7)) is 1/2 sqrt(4x^2+7)+C

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integral of (2x)/(sqrt(4x^2+7))

\int \frac{2 x}{4 x ^{2} + 7} d x

\frac{1}{2} 4 x^{2} + 7 + C

Solution steps

\int \frac{2 x}{4 x ^{2} + 7} 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}{4 x ^{2} + 7} d x

Apply u-substitution

: \int \frac{1}{8 u} d u

= 2 \cdot \int \frac{1}{8 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}{8} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= 2 \cdot \frac{1}{8} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 4 x^{2} + 7

= 2 \cdot \frac{1}{8} \cdot 2 (4 x^{2} + 7)^{\frac{1}{2}}

Simplify

2 \cdot \frac{1}{8} \cdot 2 (4 x^{2} + 7)^{\frac{1}{2}} : \frac{1}{2} 4 x^{2} + 7

= \frac{1}{2} 4 x^{2} + 7

Add a constant to the solution

= \frac{1}{2} 4 x^{2} + 7 + C

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What is the integral of (2x)/(sqrt(4x^2+7)) ?
The integral of (2x)/(sqrt(4x^2+7)) is 1/2 sqrt(4x^2+7)+C