What is the integral of (5x)/(sqrt(5x^2+4y^2)) ?

The integral of (5x)/(sqrt(5x^2+4y^2)) is sqrt(4y^2+5x^2)+C

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integral of (5x)/(sqrt(5x^2+4y^2))

\int \frac{5 x}{5 x ^{2} + 4 y ^{2}} d x

4 y^{2} + 5 x^{2} + C

Solution steps

\int \frac{5 x}{5 x ^{2} + 4 y ^{2}} d x

Take the constant out:

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

= 5 \cdot \int \frac{x}{4 y ^{2} + 5 x ^{2}} d x

Apply u-substitution

: \int \frac{1}{10 u} d u

= 5 \cdot \int \frac{1}{10 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{10} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= 5 \cdot \frac{1}{10} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 4 y^{2} + 5 x^{2}

= 5 \cdot \frac{1}{10} \cdot 2 (4 y^{2} + 5 x^{2})^{\frac{1}{2}}

Simplify

5 \cdot \frac{1}{10} \cdot 2 (4 y^{2} + 5 x^{2})^{\frac{1}{2}} : 4 y^{2} + 5 x^{2}

= 4 y^{2} + 5 x^{2}

Add a constant to the solution

= 4 y^{2} + 5 x^{2} + C

\int - 8 x^{- 9} d x

\int 0.733 x e^{- 0.27 x} d x

\int 18 x cos (3 x) d x

x y + y^{2} - x^{2} \cdot \frac{d y}{d x} = 0

What is the integral of (5x)/(sqrt(5x^2+4y^2)) ?
The integral of (5x)/(sqrt(5x^2+4y^2)) is sqrt(4y^2+5x^2)+C