What is the integral of (4x)/(sqrt(50-x^2)) ?

The integral of (4x)/(sqrt(50-x^2)) is -4sqrt(50-x^2)+C

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integral of (4x)/(sqrt(50-x^2))

\int \frac{4 x}{50 - x ^{2}} d x

- 4 50 - x^{2} + C

Solution steps

\int \frac{4 x}{50 - x ^{2}} d x

Take the constant out:

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

= 4 \cdot \int \frac{x}{50 - x ^{2}} d x

Apply u-substitution

: \int - \frac{1}{2 u} d u

= 4 \cdot \int - \frac{1}{2 u} d u

Take the constant out:

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

= 4 (- \frac{1}{2} \cdot \int \frac{1}{u} d u)

Apply the Power Rule

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

= 4 (- \frac{1}{2} \cdot 2 u^{\frac{1}{2}})

Substitute back

u = 50 - x^{2}

= 4 (- \frac{1}{2} \cdot 2 (50 - x^{2})^{\frac{1}{2}})

Simplify

4 (- \frac{1}{2} \cdot 2 (50 - x^{2})^{\frac{1}{2}}) : - 4 50 - x^{2}

= - 4 50 - x^{2}

Add a constant to the solution

= - 4 50 - x^{2} + C

arctan (x^{2})

\frac{d y}{d x} - \frac{3 x ^{2}}{2 y} = 0

\frac{d}{d x} (arctan (2 x^{4}))

\int (x^{2} + 3)^{4} 2 x d x

\int e^{- a x^{2}} d x

What is the integral of (4x)/(sqrt(50-x^2)) ?
The integral of (4x)/(sqrt(50-x^2)) is -4sqrt(50-x^2)+C