What is the integral of 1/((144+x^2)^{3/2)} ?

The integral of 1/((144+x^2)^{3/2)} is x/(144(144+x^2)^{1/2)}+C

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integral of 1/((144+x^2)^{3/2)}

\int \frac{1}{( 144 + x ^{2} ) ^{\frac{3}{2}}} d x

\frac{x}{144 ( 144 + x ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{1}{( 144 + x ^{2} ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

: \int \frac{1}{144 sec ( u )} d u

= \int \frac{1}{144 sec ( u )} d u

Take the constant out:

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

= \frac{1}{144} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{144} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{144} sin (u)

Substitute back

u = arctan (\frac{1}{12} x)

= \frac{1}{144} sin (arctan (\frac{1}{12} x))

Simplify

\frac{1}{144} sin (arctan (\frac{1}{12} x)) : \frac{x}{144 ( 144 + x ^{2} ) ^{\frac{1}{2}}}

= \frac{x}{144 ( 144 + x ^{2} ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x}{144 ( 144 + x ^{2} ) ^{\frac{1}{2}}} + C

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

\frac{d y}{d x} = y (x y^{7} - 1)

\int sin^{3} (x co) s^{9} x d x

y = - 6 x^{2} + 5 x, (0, 0)

\frac{d y}{d t} + y = cos (t), y (0) = 1

What is the integral of 1/((144+x^2)^{3/2)} ?
The integral of 1/((144+x^2)^{3/2)} is x/(144(144+x^2)^{1/2)}+C