What is the integral of (5x)/(sqrt(1-2x^2)) ?

The integral of (5x)/(sqrt(1-2x^2)) is -5/2 sqrt(1-2x^2)+C

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integral of (5x)/(sqrt(1-2x^2))

\int \frac{5 x}{1 - 2 x ^{2}} d x

- \frac{5}{2} 1 - 2 x^{2} + C

Solution steps

\int \frac{5 x}{1 - 2 x ^{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}{1 - 2 x ^{2}} d x

Apply u-substitution

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

= 5 \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

= 5 (- \frac{1}{4} \cdot \int \frac{1}{u} d u)

Apply the Power Rule

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

= 5 (- \frac{1}{4} \cdot 2 u^{\frac{1}{2}})

Substitute back

u = 1 - 2 x^{2}

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

Simplify

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

= - \frac{5}{2} 1 - 2 x^{2}

Add a constant to the solution

= - \frac{5}{2} 1 - 2 x^{2} + C

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What is the integral of (5x)/(sqrt(1-2x^2)) ?
The integral of (5x)/(sqrt(1-2x^2)) is -5/2 sqrt(1-2x^2)+C