What is the integral of 1/((361+x^2)^{3/2)} ?

The integral of 1/((361+x^2)^{3/2)} is x/(361(361+x^2)^{1/2)}+C

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integral of 1/((361+x^2)^{3/2)}

\int \frac{1}{( 361 + x ^{2} ) ^{\frac{3}{2}}} d x

\frac{x}{361 ( 361 + x ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{1}{( 361 + x ^{2} ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

: \int \frac{1}{361 sec ( u )} d u

= \int \frac{1}{361 sec ( u )} d u

Take the constant out:

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

= \frac{1}{361} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{361} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{361} sin (u)

Substitute back

u = arctan (\frac{1}{19} x)

= \frac{1}{361} sin (arctan (\frac{1}{19} x))

Simplify

\frac{1}{361} sin (arctan (\frac{1}{19} x)) : \frac{x}{361 ( 361 + x ^{2} ) ^{\frac{1}{2}}}

= \frac{x}{361 ( 361 + x ^{2} ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x}{361 ( 361 + x ^{2} ) ^{\frac{1}{2}}} + C

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

\int tan (11 x se) c^{2} 11 x d x

\frac{d P}{d t} = P (1 - P)

\frac{d}{d x} (\frac{ln ( 2 x - 1 )}{x ^{2} - 1})

What is the integral of 1/((361+x^2)^{3/2)} ?
The integral of 1/((361+x^2)^{3/2)} is x/(361(361+x^2)^{1/2)}+C