What is the integral of 1/((441+x^2)^{3/2)} ?

The integral of 1/((441+x^2)^{3/2)} is x/(441(441+x^2)^{1/2)}+C

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integral of 1/((441+x^2)^{3/2)}

\int \frac{1}{( 441 + x ^{2} ) ^{\frac{3}{2}}} d x

\frac{x}{441 ( 441 + x ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{1}{( 441 + x ^{2} ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

: \int \frac{1}{441 sec ( u )} d u

= \int \frac{1}{441 sec ( u )} d u

Take the constant out:

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

= \frac{1}{441} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{441} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{441} sin (u)

Substitute back

u = arctan (\frac{1}{21} x)

= \frac{1}{441} sin (arctan (\frac{1}{21} x))

Simplify

\frac{1}{441} sin (arctan (\frac{1}{21} x)) : \frac{x}{441 ( 441 + x ^{2} ) ^{\frac{1}{2}}}

= \frac{x}{441 ( 441 + x ^{2} ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x}{441 ( 441 + x ^{2} ) ^{\frac{1}{2}}} + C

x^{^{'}} = k (300 - x)

\frac{d}{d x} (\frac{8 x ^{3} - 9}{x ^{2}})

\int \frac{1}{u ^{5}} d u

\frac{d}{d x} (4 cos (x) - sin (x))

x^{2} y^{^{''}} - 3 x y^{^{'}} + 3 y = 0, y (1) = 5, y^{^{'}} (1) = 3

What is the integral of 1/((441+x^2)^{3/2)} ?
The integral of 1/((441+x^2)^{3/2)} is x/(441(441+x^2)^{1/2)}+C