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I am trying to integrate this:

$$\int_0^2 \frac{dt}{\sqrt{4 + t^2}}$$

after letting $t = 2 \tan {\theta}$, I find that the equation reduces to:

$$\int \sec {\theta}\, d \theta = \ln | \sec {\theta} + \tan {\theta} |$$ and I find that the indefinite integral after replacing the thetas is is:

$$\ln \Bigg| \frac{\sqrt{t^2 + 4}}{2} + t \,\Bigg| $$

when I take the definite integral I get:

$$\ln \big|\, 2 + \sqrt{2} \,\big| $$ which is not what wolfram gets:

enter image description here

What am I doing wrong?

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You made an algebra error when back substituting. It should be

$$ \ln\left|\frac{1}{2}\sqrt{t^2+4} + \frac{t}{2}\right| $$

This gives $\ln(1+\sqrt{2}) = \sinh^{-1}(1)$.

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