What is #f(x) = int (2x-xe^x)(-xe^x+secx) dx# if #f(0 ) = 5 #?
Can't be perfectly determined.
First, Integrate.
Now We have to Integrate By Parts.
Now The Problem is,
So, The Entire Integral is :
Hope this helps, but It won't, most probably.
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To find ( f(x) ), we need to integrate the given function ( (2x - xe^x)(-xe^x + \sec(x)) ) with respect to ( x ). Then, we'll use the initial condition ( f(0) = 5 ) to determine the constant of integration.
[ \begin{align*} f(x) &= \int (2x - xe^x)(-xe^x + \sec(x)) , dx \ &= \int (-2x^2e^x + xe^{2x} - 2x\sec(x) + xe^x\sec(x)) , dx \ &= -\frac{2}{3}x^3e^x + \frac{1}{2}xe^{2x} - 2x\ln|\sec(x) + \tan(x)| + \frac{1}{2}x^2 + C \end{align*} ]
Given that ( f(0) = 5 ), we can substitute ( x = 0 ) into the expression for ( f(x) ) and solve for ( C ).
[ \begin{align*} 5 &= -\frac{2}{3}(0)^3e^0 + \frac{1}{2}(0)e^{2 \cdot 0} - 2(0)\ln|\sec(0) + \tan(0)| + \frac{1}{2}(0)^2 + C \ 5 &= 0 + 0 - 0 + 0 + C \ C &= 5 \end{align*} ]
Therefore, the function ( f(x) ) is:
[ f(x) = -\frac{2}{3}x^3e^x + \frac{1}{2}xe^{2x} - 2x\ln|\sec(x) + \tan(x)| + \frac{1}{2}x^2 + 5 ]
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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