What is the integral of (3x)/(sqrt(2x^2+5)) ?

The integral of (3x)/(sqrt(2x^2+5)) is 3/2 sqrt(2x^2+5)+C

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integral of (3x)/(sqrt(2x^2+5))

\int \frac{3 x}{2 x ^{2} + 5} d x

\frac{3}{2} 2 x^{2} + 5 + C

Solution steps

\int \frac{3 x}{2 x ^{2} + 5} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{2 x ^{2} + 5} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

= 3 \cdot \int \frac{1}{4 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= 3 \cdot \frac{1}{4} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 2 x^{2} + 5

= 3 \cdot \frac{1}{4} \cdot 2 (2 x^{2} + 5)^{\frac{1}{2}}

Simplify

3 \cdot \frac{1}{4} \cdot 2 (2 x^{2} + 5)^{\frac{1}{2}} : \frac{3}{2} 2 x^{2} + 5

= \frac{3}{2} 2 x^{2} + 5

Add a constant to the solution

= \frac{3}{2} 2 x^{2} + 5 + C

\frac{\partial}{\partial x} (cos (- t + y + e^{x}))

\int x^{5} e^{x^{2}} d x

\frac{d}{d x} (ln (cos (ln (x))))

\frac{d y}{d t} = 3 t e^{- y}

y = x arcsin (2 x)

What is the integral of (3x)/(sqrt(2x^2+5)) ?
The integral of (3x)/(sqrt(2x^2+5)) is 3/2 sqrt(2x^2+5)+C