Mathematics: Integrate (2(x+1)^-1/2) -1 dx and show me how the substitution rule works. is it possible without using the substitution rule?

Integrate (2(x+1)^-1/2) -1 dx and show me how the

Ответ:

logan12345677885675

09.01.2022

$\displaystyle\int(2(x+1)^{-1/2}-1)\,\mathrm dx$

Here we would substitute y=x+1

, so that $\mathrm dy=\mathrm dx$ , and so

$\displaystyle\int(2(x+1)^{-1/2}-1)\,\mathrm dx=\int(2y^{-1/2}-1)\,\mathrm dy$

I don't really see a way around doing this with a substitution. This sort of process is involved with any integrand that includes a composite function.

Now you can integrate term-by-term with the power rule:

$\displaystyle\int(2y^{-1/2}-1)\,\mathrm dy=\dfrac{2y^{1/2}}{\frac12}-y+C$
$=4y^{1/2}-y+C$
$=4(x+1)^{1/2}-(x+1)+C$
$=4(x+1)^{1/2}-x+C$

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Ответ:

aj2869

03.09.2020

Mathematics

Anything less than 12 (must be positive numbers and it mustn’t be 0) is greater than 0.

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Integrate (2(x+1)^-1/2) -1 dx and show me how the substitution rule works. is it possible without using the substitution rule?

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