What is the integral from 1 to 2x of (t^2)/(t^2+1) ?

The integral from 1 to 2x of (t^2)/(t^2+1) is -arctan(2x)+pi/4+2x-1

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integral from 1 to 2x of (t^2)/(t^2+1)

\int_{1}^{2 x} \frac{t ^{2}}{t ^{2} + 1} d t

- arctan (2 x) + \frac{π}{4} + 2 x - 1

Solution steps

\int_{1}^{2 x} \frac{t ^{2}}{t ^{2} + 1} d t

\frac{t ^{2}}{t ^{2} + 1} = - \frac{1}{t ^{2} + 1} + 1

= \int_{1}^{2 x} - \frac{1}{t ^{2} + 1} + 1 d t

Apply the Sum Rule:

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

= - \int_{1}^{2 x} \frac{1}{t ^{2} + 1} d t + \int_{1}^{2 x} 1 d t

\int_{1}^{2 x} \frac{1}{t ^{2} + 1} d t = arctan (2 x) - \frac{π}{4}

\int_{1}^{2 x} 1 d t = 2 x - 1

= - (arctan (2 x) - \frac{π}{4}) + 2 x - 1

Simplify

= - arctan (2 x) + \frac{π}{4} + 2 x - 1

What is the integral from 1 to 2x of (t^2)/(t^2+1) ?
The integral from 1 to 2x of (t^2)/(t^2+1) is -arctan(2x)+pi/4+2x-1