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 January 26th, 2010, 04:23 PM #1 Newbie   Joined: Jan 2010 Posts: 3 Thanks: 0 Indefinite Integration How do you solve this integral? http://www.webassign.net/cgi-bin/symima ... x%20%3D%20
January 26th, 2010, 06:26 PM   #2
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Joined: Dec 2006
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Re: Indefinite Integration

Hello, bejoty1!

Straight substitution . . .

Quote:
 $\int 8x^{\frac{1}{2}}\,\left(1\,+\,x^{\frac{3}{2}}\righ t)^{10}\,dx$

$\text{We have: }\;8\int\left(1\,+\,x^{\frac{3}{2}}\right)\left(x^ {\frac{1}{2}}\,dx\right)$

$\text{Let: }\:u\:=\:1\,+\,x^{\frac{3}{2}} \;\;\;\Rightarrow\;\;\;du \:=\:\frac{3}{2}\,x^{\frac{1}{2}}\,dx \;\;\;\Rightarrow\;\;\;x^{\frac{1}{2}}\,dx \:=\:\frac{2}{3}\,du$

$\text{Substitute: }\;8\int u^{10}\,\left(\frac{2}{3}\,du\right) \;=\; \frac{16}{3}\int u^{10}\,du \;=\;\frac{16}{3}\,\frac{u^{11}}{11}\,+\,C$

$\text{Back-substitute: }\;\frac{16}{33}\,\left(1\,+\,x^{\frac{3}{2}}\righ t)^{11}\,+\,C$

 January 27th, 2010, 06:34 AM #3 Newbie   Joined: Jan 2010 Posts: 3 Thanks: 0 Re: Indefinite Integration Oh wow, I can't believe I missed pulling the 8 out in front. That definitely helped. Thank you so much!

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