What is the integral of 25arctan(sqrt(x)) ?

The integral of 25arctan(sqrt(x)) is 50(1/2 xarctan(sqrt(x))-1/2 (-arctan(sqrt(x))+sqrt(x)))+C

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integral of 25arctan(sqrt(x))

\int 25 arctan (x) d x

50 (\frac{1}{2} x arctan (x) - \frac{1}{2} (- arctan (x) + x)) + C

Solution steps

\int 25 arctan (x) d x

Take the constant out:

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

= 25 \cdot \int arctan (x) d x

Apply u-substitution

: \int arctan (u) \cdot 2 u d u

= 25 \cdot \int arctan (u) \cdot 2 u d u

Take the constant out:

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

= 25 \cdot 2 \cdot \int arctan (u) u d u

Apply Integration By Parts

: \frac{1}{2} u^{2} arctan (u) - \int \frac{u ^{2}}{2 ( u ^{2} + 1 )} d u

= 25 \cdot 2 (\frac{1}{2} u^{2} arctan (u) - \int \frac{u ^{2}}{2 ( u ^{2} + 1 )} d u)

\int \frac{u ^{2}}{2 ( u ^{2} + 1 )} d u = \frac{1}{2} (- arctan (u) + u)

= 25 \cdot 2 (\frac{1}{2} u^{2} arctan (u) - \frac{1}{2} (- arctan (u) + u))

Substitute back

u = x

= 25 \cdot 2 (\frac{1}{2} (x)^{2} arctan (x) - \frac{1}{2} (- arctan (x) + x))

Simplify

25 \cdot 2 (\frac{1}{2} (x)^{2} arctan (x) - \frac{1}{2} (- arctan (x) + x)) : 50 (\frac{1}{2} x arctan (x) - \frac{1}{2} (- arctan (x) + x))

= 50 (\frac{1}{2} x arctan (x) - \frac{1}{2} (- arctan (x) + x))

Add a constant to the solution

= 50 (\frac{1}{2} x arctan (x) - \frac{1}{2} (- arctan (x) + x)) + C

\frac{d ^{2} y}{d x ^{2}} + 20 \frac{d y}{d x} + 200 y = 300

\int_{1}^{\infty} 3 e^{- x} d x

7 x^{2} - 3 x

\int_{- \infty}^{0} \frac{1}{6 - 8 x} d x

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What is the integral of 25arctan(sqrt(x)) ?
The integral of 25arctan(sqrt(x)) is 50(1/2 xarctan(sqrt(x))-1/2 (-arctan(sqrt(x))+sqrt(x)))+C