What is the integral of (x^2)/((25-x^2)^{3/2)} ?

The integral of (x^2)/((25-x^2)^{3/2)} is -arcsin(1/5 x)+x/((25-x^2)^{1/2)}+C

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integral of (x^2)/((25-x^2)^{3/2)}

\int \frac{x ^{2}}{( 25 - x ^{2} ) ^{\frac{3}{2}}} d x

- arcsin (\frac{1}{5} x) + \frac{x}{( 25 - x ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{x ^{2}}{( 25 - x ^{2} ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

: \int tan^{2} (u) d u

= \int tan^{2} (u) d u

Rewrite using trig identities

= \int - 1 + sec^{2} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d u + \int sec^{2} (u) d u

\int 1 d u = u

\int sec^{2} (u) d u = tan (u)

= - u + tan (u)

Substitute back

u = arcsin (\frac{1}{5} x)

= - arcsin (\frac{1}{5} x) + tan (arcsin (\frac{1}{5} x))

Simplify

- arcsin (\frac{1}{5} x) + tan (arcsin (\frac{1}{5} x)) : - arcsin (\frac{1}{5} x) + \frac{x}{( 25 - x ^{2} ) ^{\frac{1}{2}}}

= - arcsin (\frac{1}{5} x) + \frac{x}{( 25 - x ^{2} ) ^{\frac{1}{2}}}

Add a constant to the solution

= - arcsin (\frac{1}{5} x) + \frac{x}{( 25 - x ^{2} ) ^{\frac{1}{2}}} + C

2 t + 1

x = 6 y^{3}, x = 21 y^{2}

\frac{d y}{d t} = \frac{4 y}{t}

\int 3 t^{- 7} d t

4 x^{\frac{5}{4}} + 2 x^{\frac{3}{2}} + 3 x

What is the integral of (x^2)/((25-x^2)^{3/2)} ?
The integral of (x^2)/((25-x^2)^{3/2)} is -arcsin(1/5 x)+x/((25-x^2)^{1/2)}+C