3u Integration Question (1 Viewer)

sarkar_as · Feb 1, 2017

Can i have some help with these questions please?

Screen Shot 2017-02-01 at 3.45.14 pm.png

Mahan1 · Feb 12, 2017

sarkar_as said:
Can i have some help with these questions please?

View attachment 33758

4a) $u = 2x+7$ implies $du = 2dx$ and
$x= \frac{u-7}{2}$
Then
$\int x\sqrt{2x+7}dx = \int (\frac{u-7}{2})\sqrt{u} (\frac{du}{2})=\frac{1}{4} \int (u-7)\sqrt{u} du = \frac{1}{4} \int (u^{\frac{3}{2}}-7u^{\frac{1}{2}}) = \frac{1}{10}u^{\frac{5}{2}} - 7\frac{1}{6}u^{\frac{3}{2}}$
To get the final answer we sub back x instead of u:
$\int x\sqrt{x}dx = \frac{1}{10}(2x+7)^{\frac{5}{2}} - 7\frac{1}{6}(2x+7)^{\frac{3}{2}}$

4b) $u^{2} = 2x+7$ implies $2u du = 2 dx$ by chain rule
$\int x\sqrt{2x+7}dx = \frac{1}{2}\int (u^{2}-7)u(udu) = \frac{1}{2}\int (u^{4}-7u^{2}) du = \frac{1}{10}u^{5} - \frac{1}{6}7u^{3}$
sub x back and you get the same answer as part a

5. if $u^{2} = x+2$ that implies $u^{2}-4 = x-2$ which means $2udu = dx$
Therefore
$\int \frac{x-2}{\sqrt{x+2}}dx = 2\int \frac{u^{2}-4}{u}udu = 2 \int u(u -\frac{4}{u}) du = 2 \int( u^{2} -4) du = \frac{2}{3}u^{3} - 8u$
sub back x to get the final answer.

pikachu975 · Feb 12, 2017

Mahan1 said:
4a) $u = 2x+7$ implies $du = 2dx$ and
$x= \frac{u-7}{2}$
Then
$\int x\sqrt{2x+7}dx = \int (\frac{u-7}{2})\sqrt{u} (\frac{du}{2})=\frac{1}{4} \int (u-7)\sqrt{u} du = \frac{1}{4} \int (u^{\frac{3}{2}}-u^{\frac{1}{2}}) = \frac{1}{10}u^{\frac{5}{2}} - \frac{1}{6}u^{\frac{3}{2}}$
To get the final answer we sub back x instead of u:
$\int x\sqrt{x}dx = \frac{1}{10}(2x+7)^{\frac{5}{2}} - \frac{1}{6}(2x+7)^{\frac{3}{2}}$

4b) $u^{2} = 2x+7$ implies $2u du = 2 dx$ by chain rule
$\int x\sqrt{2x+7}dx = \frac{1}{2}\int (u^{2}-7)u(udu) = \frac{1}{2}\int (u^{4}-7u^{2}) du = \frac{1}{10}u^{5} - \frac{1}{6}7u^{3}$
sub x back and you get the same answer as part a

5. if $u^{2} = x+2$ that implies $u^{2}-4 = x-2$ which means $2udu = dx$
Therefore
$\int \frac{x-2}{\sqrt{x+2}}dx = 2\int \frac{u^{2}-4}{u}udu = 2 \int u(u -\frac{4}{u}) du = 2 \int( u^{2} -4) du = \frac{2}{3}u^{3} - 8u$
sub back x to get the final answer.

I haven't looked through the rest of the answer but I think in part a you forgot the 7 in (u-7)*sqrt(u) when you expanded it.

Mahan1 · Feb 12, 2017

pikachu975 said:
I haven't looked through the rest of the answer but I think in part a you forgot the 7 in (u-7)*sqrt(u) when you expanded it.

Thank you for pointing that out, I fixed it.

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3u Integration Question (1 Viewer)

sarkar_as

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Mahan1

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Mahan1

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