What is the integral of-(200x)/((14+1/2 x^2)^{3/2)} ?

The integral of-(200x)/((14+1/2 x^2)^{3/2)} is (400*2^{1/2})/((28+x^2)^{1/2)}+C

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integral of-(200x)/((14+1/2 x^2)^{3/2)}

\int - \frac{200 x}{( 14 + \frac{1}{2} x ^{2} ) ^{\frac{3}{2}}} d x

\frac{400 \cdot 2 ^{\frac{1}{2}}}{( 28 + x ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

\int - \frac{200 x}{( 14 + \frac{1}{2} x ^{2} ) ^{\frac{3}{2}}} d x

Take the constant out:

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

= - 200 \cdot \int \frac{x}{( 14 + \frac{1}{2} x ^{2} ) ^{\frac{3}{2}}} d x

Simplify

\frac{x}{( 14 + \frac{1}{2} x ^{2} ) ^{\frac{3}{2}}} : \frac{2 \cdot 2 ^{\frac{1}{2}} x}{( 28 + x ^{2} ) ^{\frac{3}{2}}}

= - 200 \cdot \int \frac{2 \cdot 2 ^{\frac{1}{2}} x}{( 28 + x ^{2} ) ^{\frac{3}{2}}} d x

Take the constant out:

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

= - 200 \cdot 2 \cdot 2^{\frac{1}{2}} \cdot \int \frac{x}{( 28 + x ^{2} ) ^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{1}{2 u ^{\frac{3}{2}}} d u

= - 200 \cdot 2 \cdot 2^{\frac{1}{2}} \cdot \int \frac{1}{2 u ^{\frac{3}{2}}} d u

Take the constant out:

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

= - 200 \cdot 2 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{\frac{3}{2}}} d u

Apply the Power Rule

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

= - 200 \cdot 2 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} (- \frac{2}{u ^{\frac{1}{2}}})

Substitute back

u = 28 + x^{2}

= - 200 \cdot 2 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} (- \frac{2}{( 28 + x ^{2} ) ^{\frac{1}{2}}})

Simplify

- 200 \cdot 2 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} (- \frac{2}{( 28 + x ^{2} ) ^{\frac{1}{2}}}) : \frac{400 \cdot 2 ^{\frac{1}{2}}}{( 28 + x ^{2} ) ^{\frac{1}{2}}}

= \frac{400 \cdot 2 ^{\frac{1}{2}}}{( 28 + x ^{2} ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{400 \cdot 2 ^{\frac{1}{2}}}{( 28 + x ^{2} ) ^{\frac{1}{2}}} + C

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What is the integral of-(200x)/((14+1/2 x^2)^{3/2)} ?
The integral of-(200x)/((14+1/2 x^2)^{3/2)} is (400*2^{1/2})/((28+x^2)^{1/2)}+C