1. ## Integrals Rp_4-5

$\int_1^2 \frac{1}{(x^2-2x+5)^{3/2}}dx=$

$\int 5^{\sqrt{7x+2}}dx=$

In the secound problem i tried maybe $e^{(\sqrt{7x+2}*ln{5})}$ but I don't know how to continue...

2. Originally Posted by gilyos
$\int_1^2 \frac{1}{(x^2-2x+5)^{3/2}}dx=$

$\int {\frac{{dx}}
{{{{\left( {{x^2} - 2x + 5} \right)}^{3/2}}}}} = \int {\frac{{dx}}
{{{{\left( {{{\left( {x - 1} \right)}^2} + 4} \right)}^{3/2}}}}} = \left\{ \begin{gathered}
x - 1 = 2\tan u, \hfill \\
dx = \frac{{2du}}
{{{{\cos }^2}u}} \hfill \\
\end{gathered} \right\} =$

$= \frac{1}
{4}\int {\frac{{du}}
{{{{\cos }^2}u{{\left( {{{\tan }^2}u + 1} \right)}^{3/2}}}}} = \frac{1}
{4}\int {\frac{{du}}
{{{{\cos }^2}u{{\left( {\frac{1}
{{{{\cos }^2}u}}} \right)}^{3/2}}}}} = \frac{1}
{4}\int {\frac{{{{\cos }^3}u}}
{{{{\cos }^2}u}}\,du} =$

$= \frac{1}
{4}\int {\cos u\,du} = \frac{1}
{4}\sin u + C = \frac{1}
{4}\sin \arctan \frac{{x - 1}}
{2} + C =$

$= \frac{1}
{4}\frac{{\frac{{x - 1}}
{2}}}
{{\sqrt {1 + {{\left( {\frac{{x - 1}}
{2}} \right)}^2}} }} + C = \frac{{x - 1}}
{{4\sqrt {{x^2} - 2x + 5} }} + C.$

Finally

$\int\limits_1^2 {\frac{{dx}}
{{{{\left( {{x^2} - 2x + 5} \right)}^{3/2}}}}} = \left. {\frac{{x - 1}}
{{4\sqrt {{x^2} - 2x + 5} }}} \right|_1^2 = \frac{1}
{{4\sqrt 5 }} = \frac{{\sqrt 5 }}
{{20}}.$

3. Originally Posted by gilyos
$\int 5^{\sqrt{7x+2}}dx=$

In the secound problem i tried maybe $e^{(\sqrt{7x+2}*ln{5})}$ but I don't know how to continue...
Try this substitution $7x + 2 = u^2$, after the integration by parts:

$\int {{5^{\sqrt {7x + 2} }}\,dx} = \left\{ \begin{gathered}
7x + 2 = {u^2} \hfill \\
dx = \frac{{2u}}
{7}\,du \hfill \\
\end{gathered} \right\} = \frac{2}
{7}\int {{5^u}u\,du} = \frac{{2 \cdot {5^u}u}}
{{7\ln 5}} - \frac{2}
{{7\ln 5}}\int {{5^u}\,du} =$

$= \frac{{2 \cdot {5^u}u}}
{{7\ln 5}} - \frac{{2 \cdot {5^u}}}
{{7{{\ln }^2}5}} + C = \left( {\frac{{2u}}
{{7\ln 5}} - \frac{2}
{{7{{\ln }^2}5}}} \right){5^u} + C =$

$= \left( {\frac{{2\sqrt {7x + 2} }}
{{7\ln 5}} - \frac{2}
{{7{{\ln }^2}5}}} \right){5^{\sqrt {7x + 2} }} + C.$

4. $= \frac{1}
{4}\int {\cos u\,du} = \frac{1}
{4}\sin u + C = \frac{1}
{4}\sin \arctan \frac{{x - 1}}
{2} + C =$

How you transposed this ? from sin(arctan)

to this :

$= \frac{1}
{4}\frac{{\frac{{x - 1}}
{2}}}
{{\sqrt {1 + {{\left( {\frac{{x - 1}}
{2}} \right)}^2}} }} + C = \frac{{x - 1}}
{{4\sqrt {{x^2} - 2x + 5} }} + C.$

5. Originally Posted by gilyos
$= \frac{1}
{4}\int {\cos u\,du} = \frac{1}
{4}\sin u + C = \frac{1}
{4}\sin \arctan \frac{{x - 1}}
{2} + C =$

How you transposed this ? from sin(arctan)

to this :

$= \frac{1}
{4}\frac{{\frac{{x - 1}}
{2}}}
{{\sqrt {1 + {{\left( {\frac{{x - 1}}
{2}} \right)}^2}} }} + C = \frac{{x - 1}}
{{4\sqrt {{x^2} - 2x + 5} }} + C.$
$\sin{\theta} = \frac{\tan{\theta}}{\sqrt{1 + \tan^2{\theta}}}$.

6. Originally Posted by gilyos
$= \frac{1}
{4}\int {\cos u\,du} = \frac{1}
{4}\sin u + C = \frac{1}
{4}\sin \arctan \frac{{x - 1}}
{2} + C =$

How you transposed this ? from sin(arctan)

to this :

$= \frac{1}
{4}\frac{{\frac{{x - 1}}
{2}}}
{{\sqrt {1 + {{\left( {\frac{{x - 1}}
{2}} \right)}^2}} }} + C = \frac{{x - 1}}
{{4\sqrt {{x^2} - 2x + 5} }} + C.$
Treat $\vartheta=\arctan\frac{x-1}{2}$. Construct a triangle from this and then find $\sin\vartheta$.

You should then see that $\sin\arctan\frac{x-1}{2}=\frac{x-1}{\sqrt{(x-1)^2+4}}=\frac{x-1}{\sqrt{x^2-2x+5}}$ as DeMath stated.

7. Originally Posted by Chris L T521
Treat $\vartheta=\arctan\frac{x-1}{2}$. Construct a triangle from this and then find $\sin\vartheta$.

You should then see that $\sin\arctan\frac{x-1}{2}=\frac{x-1}{\sqrt{(x-1)^2+4}}=\frac{x-1}{\sqrt{x^2-2x+5}}$ as DeMath stated.

So , if i get it right

$arcsin\frac{1}{2} = \frac{1}{\sqrt{1^2+2^2}} - \frac{1}{\sqrt{5}}$

?

8. Originally Posted by gilyos
So , if i get it right

$arcsin\frac{1}{2} = \frac{1}{\sqrt{1^2+2^2}} - \frac{1}{\sqrt{5}}$

?
See my post:

$\sin{\theta} = \frac{\tan{\theta}}{\sqrt{1 + \tan^2{\theta}}}$.

So if $\theta = \arctan{\frac{x - 1}{2}}$ then

$\sin{\arctan{\frac{x - 1}{2}}} = \frac{\tan{\arctan{\frac{x - 1}{2}}}}{\sqrt{1 + \left(\tan{\arctan{\frac{x - 1}{2}}}\right)^2}}$

$= \frac{\frac{x - 1}{2}}{\sqrt{1 + \left(\frac{x - 1}{2}\right)^2}}$

$= \frac{\frac{x - 1}{2}}{\sqrt{\frac{4 + (x - 1)^2}{4}}}$

$= \frac{\frac{x - 1}{2}}{\frac{\sqrt{4 + x^2 - 2x + 1}}{2}}$

$= \frac{x - 1}{\sqrt{x^2 - 2x + 5}}$.