Evaluate the definite integrals

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December 14th 2010, 07:11 AM
watp

Evaluate the definite integrals

I have absolutely no idea how I'm supposed to do this.

Evaluate each of the following definite integrals:

a) $\int_1^4 \frac{x^2+2x}{\sqrt{x^2}} dx$
b) $\int_{-1}^{1} \frac{1}{v}(3v^3-v^\frac{-1}{3}) dv$

Can anyone point me in the right direction? I've done definite integrals, but none that hard before.

Thanks in advance :)
December 14th 2010, 09:24 AM
watp

Worked out how to do it.

You integrate it first, then multiply it by the bigger number and minus it by the smaller number times the value.
December 14th 2010, 05:52 PM
Prove It

The easiest way is to simplify the integrand...

In the region $\displaystyle 1 \leq x \leq 4, \sqrt{x^2} = x$ .

So $\displaystyle \int_1^4{\frac{x^2+2x}{\sqrt{x^2}}\,dx} = \int_1^4{\frac{x^2+2x}{x}\,dx} = \int_1^4{x + 2\,dx}$ .

Also $\displaystyle \int_{-1}^1{\frac{1}{v}(3v^3 - v^{-\frac{1}{3}})\,dv} = \int_{-1}^1{3v^2 - v^{-\frac{4}{3}}\,dv}$ .

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