What is the integral of 1/((484+x^2)^{3/2)} ?

The integral of 1/((484+x^2)^{3/2)} is x/(484(484+x^2)^{1/2)}+C

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integral of 1/((484+x^2)^{3/2)}

\int \frac{1}{( 484 + x ^{2} ) ^{\frac{3}{2}}} d x

\frac{x}{484 ( 484 + x ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{1}{( 484 + x ^{2} ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

: \int \frac{1}{484 sec ( u )} d u

= \int \frac{1}{484 sec ( u )} d u

Take the constant out:

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

= \frac{1}{484} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{484} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{484} sin (u)

Substitute back

u = arctan (\frac{1}{22} x)

= \frac{1}{484} sin (arctan (\frac{1}{22} x))

Simplify

\frac{1}{484} sin (arctan (\frac{1}{22} x)) : \frac{x}{484 ( 484 + x ^{2} ) ^{\frac{1}{2}}}

= \frac{x}{484 ( 484 + x ^{2} ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x}{484 ( 484 + x ^{2} ) ^{\frac{1}{2}}} + C

\frac{3 ^{2}}{( s + 3 ) ^{2}} \cdot \frac{1}{s}

\int \frac{1 - w}{w} d w

f (x) = sin^{2} (4 x)

\frac{\partial}{\partial v} (2 sin (u) cos (v))

e^{x}, \frac{e ^{2 x}}{2}, 0, 1

What is the integral of 1/((484+x^2)^{3/2)} ?
The integral of 1/((484+x^2)^{3/2)} is x/(484(484+x^2)^{1/2)}+C