What is the integral of (250)/((x^2+25)^2) ?

The integral of (250)/((x^2+25)^2) is arctan(x/5)+1/2 sin(2arctan(x/5))+C

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integral of (250)/((x^2+25)^2)

\int \frac{250}{( x ^{2} + 25 ) ^{2}} d x

arctan (\frac{x}{5}) + \frac{1}{2} sin (2 arctan (\frac{x}{5})) + C

Solution steps

\int \frac{250}{( x ^{2} + 25 ) ^{2}} d x

Take the constant out:

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

= 250 \cdot \int \frac{1}{( x ^{2} + 25 ) ^{2}} d x

Apply Integral Substitution

= 250 \cdot \int \frac{1}{125 ( u ^{2} + 1 ) ^{2}} d u

Take the constant out:

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

= 250 \cdot \frac{1}{125} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Apply Trigonometric Substitution

: \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

= 250 \cdot \frac{1}{125} \cdot \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

Rewrite using trig identities

= 250 \cdot \frac{1}{125} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

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

= 250 \cdot \frac{1}{125} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 250 \cdot \frac{1}{125} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= 250 \cdot \frac{1}{125} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 250 \cdot \frac{1}{125} \cdot \frac{1}{2} (arctan (\frac{x}{5}) + \frac{1}{2} sin (2 arctan (\frac{x}{5})))

Simplify

250 \cdot \frac{1}{125} \cdot \frac{1}{2} (arctan (\frac{x}{5}) + \frac{1}{2} sin (2 arctan (\frac{x}{5}))) : arctan (\frac{x}{5}) + \frac{1}{2} sin (2 arctan (\frac{x}{5}))

= arctan (\frac{x}{5}) + \frac{1}{2} sin (2 arctan (\frac{x}{5}))

Add a constant to the solution

= arctan (\frac{x}{5}) + \frac{1}{2} sin (2 arctan (\frac{x}{5})) + C

y = e^{s i n (x)}

\frac{s - 9}{2 s ^{2} + s + 4}

k = 0 \sum \infty (- 1)^{k}

\frac{c ^{3}}{100} - \frac{1500}{c}

y = 25 - x^{2}, y = 9

What is the integral of (250)/((x^2+25)^2) ?
The integral of (250)/((x^2+25)^2) is arctan(x/5)+1/2 sin(2arctan(x/5))+C