Question

Antideriv question:
how can i find the anti deriv of 4/(1+4t^2) dt ?

Answer 1

The anti-derivative is the indefinite integral.

You should be asking yourself: whose derivative is of the form? \[ \frac{1}{1+x^2} \]
Hopefully you know that \[\frac{d}{dx}\arctan(x) = \frac{1}{1+x^2} \].
If I change the argument of the arctangent function from 'x' to '2t', then:
\[ \frac{d}{dt}\arctan(4t) = \frac{2}{1+(2t)^2} = \frac{2}{1+4t^2} \]
The '2' in the numerator comes from the chain rule.
But, we want a 4 in the numerator, so we'll multiply the entire equation by '2':
\[ \frac{d}{dt}2\arctan(4t) = \frac{4}{1+4t^2} \]
Of course, there is also the added constant of integration.

This is one way to do it, or you can use a trigonometric substitution to find the integral.

Answer 2

Oh sorry, the argument in the arctangent should be (2t), NOT (4t).