- #1
DaalChawal
- 85
- 0
MHB Integration Doubt: Answers & Solutions
- Thread starter DaalChawal
- Start date
-
- Tags
- Doubt Integration
Click For Summary
The discussion focuses on simplifying a complex integrand involving logarithmic and exponential functions. A proposed simplification reveals a clearer structure, making it easier to analyze. Despite efforts to apply differentiation and substitution methods, further progress in solving the integral remains elusive. Participants express uncertainty about the next steps in the integration process. The conversation highlights the challenges of tackling intricate mathematical expressions effectively.
- #2
Prove It
Gold Member
MHB
- 1,434
- 20
It might be worth simplifying the integrand...
$\displaystyle \begin{align*} \frac{\ln{\left( \mathrm{e}\,x^{x+1} \right)} + \left[ \ln{ \left( x^{\sqrt{x}} \right) } \right]^2 }{1 + x\ln{ \left( x \right) } \ln{ \left( \mathrm{e}^x\,x^x \right) }} &= \frac{ \ln{\left( \mathrm{e} \right) } + \ln{ \left( x^{x+1} \right) } + \left[ \sqrt{x} \, \ln{ \left( x \right) } \right] ^2 }{ 1 + x\ln{ \left( x \right) } \left[ \ln{\left( \mathrm{e}^x \right) } + \ln{ \left( x^x \right) } \right] } \\ &= \frac{ 1 + \left( x + 1 \right) \ln{ \left( x \right) } + x \, \left[ \ln{ \left( x \right) } \right] ^2 }{ 1 + x \ln{ \left( x \right) } \left[ x + x\ln{ \left( x \right) } \right] } \end{align*}$
I don't know if this helps, but it looks simpler at least...
$\displaystyle \begin{align*} \frac{\ln{\left( \mathrm{e}\,x^{x+1} \right)} + \left[ \ln{ \left( x^{\sqrt{x}} \right) } \right]^2 }{1 + x\ln{ \left( x \right) } \ln{ \left( \mathrm{e}^x\,x^x \right) }} &= \frac{ \ln{\left( \mathrm{e} \right) } + \ln{ \left( x^{x+1} \right) } + \left[ \sqrt{x} \, \ln{ \left( x \right) } \right] ^2 }{ 1 + x\ln{ \left( x \right) } \left[ \ln{\left( \mathrm{e}^x \right) } + \ln{ \left( x^x \right) } \right] } \\ &= \frac{ 1 + \left( x + 1 \right) \ln{ \left( x \right) } + x \, \left[ \ln{ \left( x \right) } \right] ^2 }{ 1 + x \ln{ \left( x \right) } \left[ x + x\ln{ \left( x \right) } \right] } \end{align*}$
I don't know if this helps, but it looks simpler at least...
- #3
DaalChawal
- 85
- 0
I also reached this step I tried to create differentiation inside and used substitution too but could not solve it further.
Hello!
I tried to calculate projections of curl using rotation of coordinate system but I encountered with following problem.
Given:
##rot_xA=\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}=0##
##rot_yA=\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}=1##
##rot_zA=\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}=0##
I rotated ##yz##-plane of this coordinate system by an angle ##45## degrees about ##x##-axis and used rotation matrix to...
Similar threads
- · Replies 10 ·
- · Replies 4 ·
- · Replies 17 ·
- · Replies 6 ·
- · Replies 10 ·
- · Replies 8 ·
- · Replies 23 ·
- · Replies 4 ·
- · Replies 2 ·