What is the integral of 1/((576+x^2)^{3/2)} ?

The integral of 1/((576+x^2)^{3/2)} is x/(576(576+x^2)^{1/2)}+C

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integral of 1/((576+x^2)^{3/2)}

\int \frac{1}{( 576 + x ^{2} ) ^{\frac{3}{2}}} d x

\frac{x}{576 ( 576 + x ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{1}{( 576 + x ^{2} ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

: \int \frac{1}{576 sec ( u )} d u

= \int \frac{1}{576 sec ( u )} d u

Take the constant out:

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

= \frac{1}{576} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{576} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{576} sin (u)

Substitute back

u = arctan (\frac{1}{24} x)

= \frac{1}{576} sin (arctan (\frac{1}{24} x))

Simplify

\frac{1}{576} sin (arctan (\frac{1}{24} x)) : \frac{x}{576 ( 576 + x ^{2} ) ^{\frac{1}{2}}}

= \frac{x}{576 ( 576 + x ^{2} ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x}{576 ( 576 + x ^{2} ) ^{\frac{1}{2}}} + C

\int 2 a cos (\frac{θ}{2}) d θ

\frac{d}{d x} (\frac{1}{2} x + 2)

(1 + cos^{2} (x))^{6}

t \to π lim (- 2 sin (- \frac{t}{2}))

\int_{0}^{\infty} a f (x) d x

What is the integral of 1/((576+x^2)^{3/2)} ?
The integral of 1/((576+x^2)^{3/2)} is x/(576(576+x^2)^{1/2)}+C