Verification

Erik Lehnsherr

New member
Joined
Jun 13, 2010
Messages
11
Are these correct? Did I miss anything?


1.Prove that  \displaystyle \text{1.\,Prove that}\,\, 02x2(x2+4)dx=62ln(3+22)\displaystyle \int_0^2 x^2 \sqrt{(x^2 + 4)}\,dx = 6\sqrt{2} - \ln(3 + 2\sqrt{2})

u=x2+4         u4=x2\displaystyle u = x^2 + 4\,\,\,\,\,\,\,\,\,u - 4 = x^2

\(\displaystyle = \int \frac{(u - 4)u^\frac{1}{2}}{2(u - 4)^\frac{1}{2}}\,\dot\,du\,\,\,\,\,\,\,\,\,du = 2x\,dx\)
                      dudx=dx\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{du}{dx} = dx

\(\displaystyle = \frac{1}{2}\int\frac{ (u - 4)^1}{u - 4^\frac{1}{2}}\,\dot\,\,u^\frac{1}{2}\)

=12u324u12(u4)\displaystyle = \frac{1}{2} \int \frac{u^\frac{3}{2} - 4u^\frac{1}{2}}{(u - 4)}




2.  Show that7107x25x6  dx=ln(3211)\displaystyle \text{2.\,\,Show that} \int_7^{10} \frac{7}{x^2 - 5x - 6}\,\,dx = \ln\bigg(\frac{32}{11}\bigg)
7107x25x6  dx=A(x6)+B(x+1)\displaystyle \int_7^{10} \frac{7}{x^2 - 5x - 6}\,\,dx = \frac{A}{(x - 6)} + \frac{B}{(x + 1)}

Ax+A+Bx6B\displaystyle Ax + A + Bx - 6B
A+B=0,A6b=7\displaystyle A + B = 0, A - 6b = 7
A=B     (B)6B=7\displaystyle A = -B\,\,\,\,\,(-B) - 6B = 7
A=1     7B=7   B=1\displaystyle \therefore A = 1\,\,\,\,\,-7B = 7\,\,\,B = -1

1(x6)+1x+1\displaystyle \frac{1}{(x - 6)} + \frac{1}{x + 1}

[lnx6lnx+1]710+C\displaystyle \bigg[\ln|x - 6| - \ln|x + 1|\bigg]_7^{10} + C

lnx6x+1lnx6x+1\displaystyle \ln |\frac{x - 6}{x + 1}| - \ln |\frac{x - 6}{x + 1}|
ln411ln18\displaystyle \ln |\frac{4}{11}| - \ln |\frac{1}{8}|

ln(41118)\displaystyle \ln \bigg(\frac{\frac{4}{11}}{\frac{1}{8}}\bigg)

ln(411×8)=ln(3211)\displaystyle \ln\bigg(\frac{4}{11} \times 8\bigg) = \ln \bigg(\frac{32}{11}\bigg)
 
I have trouble believing that that much work doesn't result in a little confidence.

1) Does't quite appear to be done.

2a) Your second equation is entirely incorrect. Where did the integral go? Integral = A/stuff + B/stuff
2b) "+C" is not correct.
2c) Why did you write this wonderfully long name for zero (0)? ln|(x-6)/(x+1)| - ln|(x-6)/(x+1)|

Think about what you are doing. Remember basic rules.

1) NEVER write one of these "=" unless you mean it.
2) Don't write things that are meaningless.
3) If you are working with just part of a problem, please label it so.

Good work. Clean them up and finish.
 
Erik Lehnsherr said:
Are these correct? Did I miss anything?


1.Prove that  \displaystyle \text{1.\,Prove that}\,\, 02x2(x2+4)dx=62ln(3+22)\displaystyle \int_0^2 x^2 \sqrt{(x^2 + 4)}\,dx = 6\sqrt{2} - \ln(3 + 2\sqrt{2})

u=x2+4         u4=x2\displaystyle u = x^2 + 4\,\,\,\,\,\,\,\,\,u - 4 = x^2

\(\displaystyle = \int \frac{(u - 4)u^\frac{1}{2}}{2(u - 4)^\frac{1}{2}}\,\dot\,du\,\,\,\,\,\,\,\,\,du = 2x\,dx\)


\(\displaystyle = \frac{1}{2}\int\frac{ (u - 4)^1}{(u - 4)^\frac{1}{2}}\,\dot\,\,u^\frac{1}{2}du\)

=12u324u12(u4)du\displaystyle = \frac{1}{2} \int \frac{u^\frac{3}{2} - 4u^\frac{1}{2}}{(u - 4)}du




2.  Show that7107x25x6  dx=ln(3211)\displaystyle \text{2.\,\,Show that} \int_7^{10} \frac{7}{x^2 - 5x - 6}\,\,dx = \ln\bigg(\frac{32}{11}\bigg)

7x25x6  =A(x6)+B(x+1)\displaystyle \frac{7}{x^2 - 5x - 6}\,\, = \frac{A}{(x - 6)} + \frac{B}{(x + 1)}

Ax+A+Bx6B\displaystyle Ax + A + Bx - 6B
A+B=0,A6b=7\displaystyle A + B = 0, A - 6b = 7
A=B     (B)6B=7\displaystyle A = -B\,\,\,\,\,(-B) - 6B = 7
A=1     7B=7   B=1\displaystyle \therefore A = 1\,\,\,\,\,-7B = 7\,\,\,B = -1

1(x6)1x+1\displaystyle \frac{1}{(x - 6)} - \frac{1}{x + 1}

710[1x61x+1]dx=\displaystyle \int_7^{10}\bigg[ \frac{1}{x - 6} - \frac{1}{x + 1}\bigg]dx =

[lnx6lnx+1]710=\displaystyle \bigg[\ln|x - 6| - \ln|x + 1|\bigg]_7^{10} =

ln10610+1ln767+1=\displaystyle \ln \bigg|\frac{10 - 6}{10 + 1}\bigg| - \ln \bigg|\frac{7 - 6}{7 + 1}\bigg|=

ln411ln18=\displaystyle \ln \bigg|\frac{4}{11}\bigg| - \ln \bigg|\frac{1}{8}\bigg|=

ln(41118)=\displaystyle \ln \bigg(\frac{\frac{4}{11}}{\frac{1}{8}}\bigg)=

ln(41181)=\displaystyle \ln\bigg(\frac{4}{11} \cdot \frac{8}{1}\bigg) =

ln(3211)\displaystyle \ln \bigg(\frac{32}{11}\bigg)


The above are more revisions, such as deletions, adding parentheses/"dx,"
adding in equal signs, and changing other symbols.
 
lookagain said:
Erik Lehnsherr said:
Are these correct? Did I miss anything?


1.Prove that  \displaystyle \text{1.\,Prove that}\,\, 02x2(x2+4)dx=62ln(3+22)\displaystyle \int_0^2 x^2 \sqrt{(x^2 + 4)}\,dx = 6\sqrt{2} - \ln(3 + 2\sqrt{2})

u=x2+4         u4=x2\displaystyle u = x^2 + 4\,\,\,\,\,\,\,\,\,u - 4 = x^2

\(\displaystyle = \int \frac{(u - 4)u^\frac{1}{2}}{2(u - 4)^\frac{1}{2}}\,\dot\,du\,\,\,\,\,\,\,\,\,du = 2x\,dx\)


\(\displaystyle = \frac{1}{2}\int\frac{ (u - 4)^1}{(u - 4)^\frac{1}{2}}\,\dot\,\,u^\frac{1}{2}du\)

=12u324u12(u4)du\displaystyle = \frac{1}{2} \int \frac{u^\frac{3}{2} - 4u^\frac{1}{2}}{(u - 4)}du



2.  Show that7107x25x6  dx=ln(3211)\displaystyle \text{2.\,\,Show that} \int_7^{10} \frac{7}{x^2 - 5x - 6}\,\,dx = \ln\bigg(\frac{32}{11}\bigg)

7x25x6  =A(x6)+B(x+1)\displaystyle \frac{7}{x^2 - 5x - 6}\,\, = \frac{A}{(x - 6)} + \frac{B}{(x + 1)}

Ax+A+Bx6B\displaystyle Ax + A + Bx - 6B
A+B=0,A6b=7\displaystyle A + B = 0, A - 6b = 7
A=B     (B)6B=7\displaystyle A = -B\,\,\,\,\,(-B) - 6B = 7
A=1     7B=7   B=1\displaystyle \therefore A = 1\,\,\,\,\,-7B = 7\,\,\,B = -1

1(x6)1x+1\displaystyle \frac{1}{(x - 6)} - \frac{1}{x + 1}

710[1x61x+1]dx=\displaystyle \int_7^{10}\bigg[ \frac{1}{x - 6} - \frac{1}{x + 1}\bigg]dx =

[lnx6lnx+1]710=\displaystyle \bigg[\ln|x - 6| - \ln|x + 1|\bigg]_7^{10} =

ln10610+1ln767+1=\displaystyle \ln \bigg|\frac{10 - 6}{10 + 1}\bigg| - \ln \bigg|\frac{7 - 6}{7 + 1}\bigg|=

ln411ln18=\displaystyle \ln \bigg|\frac{4}{11}\bigg| - \ln \bigg|\frac{1}{8}\bigg|=

ln(41118)=\displaystyle \ln \bigg(\frac{\frac{4}{11}}{\frac{1}{8}}\bigg)=

ln(41181)=\displaystyle \ln\bigg(\frac{4}{11} \cdot \frac{8}{1}\bigg) =

ln(3211)\displaystyle \ln \bigg(\frac{32}{11}\bigg)


The above are more revisions, such as deletions, adding parentheses/"dx,"
adding in equal signs, and changing other symbols.

Thank you.
 
Top