What is the integral from 0 to 3 of (arctan(x))/2 ?

The integral from 0 to 3 of (arctan(x))/2 is 1/2 (3arctan(3)-1/2 ln(10))

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integral from 0 to 3 of (arctan(x))/2

\int_{0}^{3} \frac{arctan ( x )}{2} d x

\frac{1}{2} (3 arctan (3) - \frac{1}{2} ln (10))

Decimal

1.29792 \dots

Solution steps

\int_{0}^{3} \frac{arctan ( x )}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{3} arctan (x) d x

Apply Integration By Parts

: [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{0}^{3}

= \frac{1}{2} [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{0}^{3}

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

= \frac{1}{2} [x arctan (x) - \frac{1}{2} ln x^{2} + 1]_{0}^{3}

Compute the boundaries:

3 arctan (3) - \frac{1}{2} ln (10)

= \frac{1}{2} (3 arctan (3) - \frac{1}{2} ln (10))

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

\int_{- c}^{c} (c - x^{2}) - (x^{2} - c) d x

\int_{0}^{3} x ln (x) d x

\int_{0}^{2} \frac{x ^{2}}{x ^{3} + 4} d x

\int_{- 4}^{2} 1

What is the integral from 0 to 3 of (arctan(x))/2 ?
The integral from 0 to 3 of (arctan(x))/2 is 1/2 (3arctan(3)-1/2 ln(10))