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$\int_0 ^\sqrt5 \frac{4x}{\sqrt{x^2+4}}dx$

I got $4(5^\frac{1}{4})$ but I'm not sure if thats right

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    Based on my calculation, it is 4.2012-07-26

3 Answers 3

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Your computation is incorrect.

The substitution $u=x^2+4$ gives $du = 2x\,dx$. When $x=0$, we have $u=4$; when $x=\sqrt{5}$, we get $u=9$. So $\begin{align*} \int_0^{\sqrt{5}}\frac{4x}{\sqrt{x^2+4}}\,dx &= 2\int_0^{\sqrt{5}}\frac{2x\,dx}{\sqrt{x^2+4}}\\ &= 2\int_4^9\frac{du}{\sqrt{u}}\\ &= 2 \int_4^9 u^{-1/2}\,du\\ &= 4u^{1/2}\Bigm|_4^9\\ &= 4(9^{1/2} - 4^{1/2})\\ &= 4(3-2)\\ &= 4. \end{align*}$

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    @Kudla69: And we could have told you as much if you had shown your work instead of just timidly putting forth your answer.2012-07-26
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There are two approaches to the definite integral when you use substitution.

$1$. Use substitution as usual, in this case $u=x^2+4$ (or maybe better $u^2=x^2+4$), to find that one indefinite integral is $4u^{1/2}$. Then replace $u$ by $x^2+4$ to find that $4(x^2+4)^{1/2}$ is an indefinite integral of your original function, and "plug in" as usual.

$2$. Use substitution, but substitute for everything. So our original integral is $\int_{x=0}^{\sqrt{5}}\frac{4x\,dx}{\sqrt{x^2+4}}.$ Note that I put in the $x$ in the limits of integration to remind myself that there is an $x$ hidden there.

Substitute, remembering to substitute for everything. So we get $\int_{u=2}^3 2u^{-1/2}\,du.$ Now the substitution has been fully done. Integrate. We get $4u^{1/2}$. Do the plugging in. We get $4$.

In general I prefer the second approach. Use one or the other, and not, as you did, a hybrid of both.

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Since it's an elementary integral, I'd solve it directly:

$I=4\int_{0}^{\sqrt{5}} \frac{x}{\sqrt{4+x^2}} dx=4\int_{0}^{\sqrt{5}} \left({\sqrt{4+x^2}}\right)' dx = 4 \sqrt{4+x^2}\Bigm|_0^{\sqrt{5}}=4.$

Q.E.D.