What is the integral from 0 to 1.5 of 4arctan(x) ?

The integral from 0 to 1.5 of 4arctan(x) is 3.53945…

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integral from 0 to 1.5 of 4arctan(x)

\int_{0}^{1.5} 4 arctan (x) d x

3.53945 \dots

Solution steps

\int_{0}^{1.5} 4 arctan (x) d x

Take the constant out:

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

= 4 \cdot \int_{0}^{1.5} arctan (x) d x

Apply Integration By Parts

= 4 [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{0}^{1.5}

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

= 4 [x arctan (x) - \frac{1}{2} ln x^{2} + 1]_{0}^{1.5}

Simplify

4 [x arctan (x) - \frac{1}{2} ln x^{2} + 1]_{0}^{1.5} : 4 [\frac{1}{2} (2 x arctan (x) - ln x^{2} + 1)]_{0}^{1.5}

= 4 [\frac{1}{2} (2 x arctan (x) - ln x^{2} + 1)]_{0}^{1.5}

Compute the boundaries:

0.88486 \dots

= 4 \cdot 0.88486 \dots

Simplify

= 3.53945 \dots

What is the integral from 0 to 1.5 of 4arctan(x) ?
The integral from 0 to 1.5 of 4arctan(x) is 3.53945…