Relationship between a^x and log to the base a, integration.

**Boz** · November 24th 2010, 03:12 PM

Could you please help me with the following problem:

Describe the relationship between $a^x$ and log to the base a, giving the exponential function as an example. Simplify the function
$\frac{e^{sin2x} lne^{3x}}{{(e^{cosx})}^{2sinx}e^{-x}}$

and hence evaluate the integral

$\int^1_0 \frac{e^{sin2x} lne^{3x}}{{(e^{cosx})}^{2sinx}e^{-x}}dx$

Thank you in advance

**skeeter** · November 24th 2010, 03:19 PM

Originally Posted by Boz

Could you please help me with the following problem:

Describe the relationship between $a^x$ and log to the base a, giving the exponential function as an example. Simplify the function
$\frac{e^{sin2x} lne^{3x}}{{(e^{cosx})}^{2sinx}e^{-x}}$

and hence evaluate the integral

$\int^1_0 \frac{e^{sin2x} lne^{3x}}{{(e^{cosx})}^{2sinx}e^{-x}}dx$

Thank you in advance

note that ...

$\sin(2x) = 2\sin{x}\cos{x}$

$\ln(e^{3x}) = 3x$ i.e. ... $\log_a(a^x) = x$

you will end up with the definite integral $\displaystyle \int_0^1 3x \cdot e^x \, dx$ which can be evaluated using integration by parts.

**Plato** · November 24th 2010, 03:20 PM

Originally Posted by Boz

Could you please help me with the following problem:

Describe the relationship between $\displaystyle a^x$ and log to the base a, giving the exponential function as an example. Simplify the function
$\displaystyle \frac{e^{sin2x} lne^{3x}}{{(e^{cosx})}^{2sinx}e^{-x}}$

and hence evaluate the integral

$\displaystyle \int^1_0 \frac{e^{sin2x} lne^{3x}}{{(e^{cosx})}^{2sinx}e^{-x}}dx$

Is that not easier to read?
Please learn to use \displaystyle

**HallsofIvy** · November 25th 2010, 01:52 AM

Originally Posted by Boz

Could you please help me with the following problem:

Describe the relationship between $a^x$ and log to the base a, giving the exponential function as an example.

$log_a(x)$ and $a^x$ are inverse functions: $log_a(a^x)= x$ and $a^{log_a(x)}= x$

Simplify the function
$\frac{e^{sin2x} lne^{3x}}{{(e^{cosx})}^{2sinx}e^{-x}}$

$ln(e^3x)= 3x$ , $(e^{cos x})^{2sin x}= e^{2sin x cos x}= e^{sin 2x}$ so $\frac{e^{sin 2x}lne^{3x}}{(e^{cosx})^{2sinx}e^{-x}}= \frac{e^{sin 2x}3x}{e^{sin 2x}}e^{-x}= 3xe^{-x}$

Integrate using integration by parts: $u= e^{-x}$ , $dv= 3x$ .

and hence evaluate the integral

$\int^1_0 \frac{e^{sin2x} lne^{3x}}{{(e^{cosx})}^{2sinx}e^{-x}}dx$

Thank you in advance

[/QUOTE]

**Boz** · November 26th 2010, 05:15 AM

Originally Posted by hallsofivy

$log_a(x)$ and $a^x$ are inverse functions: $log_a(a^x)= x$ and $a^{log_a(x)}= x$

$ln(e^3x)= 3x$ , $(e^{cos x})^{2sin x}= e^{2sin x cos x}= e^{sin 2x}$ so $\frac{e^{sin 2x}lne^{3x}}{(e^{cosx})^{2sinx}e^{-x}}= \frac{e^{sin 2x}3x}{e^{sin 2x}}e^{-x}= 3xe^{-x}$

integrate using integration by parts: $u= e^{-x}$ , $dv= 3x$ .

And hence evaluate the integral

$\int^1_0 \frac{e^{sin2x} lne^{3x}}{{(e^{cosx})}^{2sinx}e^{-x}}dx$

thank you in advance

[/quote]

Here is what I did. Is it correct?

$\displaystyle \int 3xe^x dx$

Let $u=3x, \frac{du}{dx}=3,$ $du=3dx,<br /> Let ,dv=e^x dx, v=\int e^x dx=e^x$
$\displaystyle 3xe^x-\int e^x 3dx,<br /> 3xe^x-3e^x=3e^x(x-1)$
Hence $\displaystyle \int_0^1 3xe^x dx= [3e^x(x-1)]<br /> 0 to 1= [3e^1(1-1)]-[3e^0(0-1)]=-3*(-1)=3$

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