Integral of e^(-x)cos(nx) dx
The solution
Detail solution
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=cos(nx) and let dv(x)=e−x.
Then du(x)=−nsin(nx).
To find v(x):
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There are multiple ways to do this integral.
Method #1
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Let u=−x.
Then let du=−dx and substitute −du:
∫eudu
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The integral of a constant times a function is the constant times the integral of the function:
∫(−eu)du=−∫eudu
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The integral of the exponential function is itself.
∫eudu=eu
So, the result is: −eu
Now substitute u back in:
Method #2
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Let u=e−x.
Then let du=−e−xdx and substitute −du:
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The integral of a constant times a function is the constant times the integral of the function:
∫(−1)du=−∫1du
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The integral of a constant is the constant times the variable of integration:
∫1du=u
So, the result is: −u
Now substitute u back in:
Now evaluate the sub-integral.
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The integral of a constant times a function is the constant times the integral of the function:
∫ne−xsin(nx)dx=n∫e−xsin(nx)dx
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=sin(nx) and let dv(x)=e−x.
Then du(x)=ncos(nx).
To find v(x):
-
Let u=−x.
Then let du=−dx and substitute −du:
∫eudu
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−eu)du=−∫eudu
-
The integral of the exponential function is itself.
∫eudu=eu
So, the result is: −eu
Now substitute u back in:
Now evaluate the sub-integral.
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The integral of a constant times a function is the constant times the integral of the function:
∫(−ne−xcos(nx))dx=−n∫e−xcos(nx)dx
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Don't know the steps in finding this integral.
But the integral is
{2xe−xsinh(x)+2xe−xcosh(x)−2e−xcosh(x)n2ex+exnsin(nx)−n2ex+excos(nx)forn=−i∨n=iotherwise
So, the result is: −n({2xe−xsinh(x)+2xe−xcosh(x)−2e−xcosh(x)n2ex+exnsin(nx)−n2ex+excos(nx)forn=−i∨n=iotherwise)
So, the result is: n(n({2xe−xsinh(x)+2xe−xcosh(x)−2e−xcosh(x)n2ex+exnsin(nx)−n2ex+excos(nx)forn=−i∨n=iotherwise)−e−xsin(nx))
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Now simplify:
{−(2n2(xsinh(x)+xcosh(x)−cosh(x))−nsin(nx)+cos(nx))e−xn2+1(nsin(nx)−cos(nx))e−xforn=−i∨n=iotherwise
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Add the constant of integration:
{−(2n2(xsinh(x)+xcosh(x)−cosh(x))−nsin(nx)+cos(nx))e−xn2+1(nsin(nx)−cos(nx))e−xforn=−i∨n=iotherwise+constant
The answer is:
{−(2n2(xsinh(x)+xcosh(x)−cosh(x))−nsin(nx)+cos(nx))e−xn2+1(nsin(nx)−cos(nx))e−xforn=−i∨n=iotherwise+constant
The answer (Indefinite)
[src]
/ // -x -x -x \ \
/ | || cosh(x)*e x*cosh(x)*e x*e *sinh(x) | |
| | ||- ----------- + ------------- + ------------- for Or(n = -I, n = I)| |
| -x | || 2 2 2 | -x | -x
| e *cos(n*x) dx = C - n*|n*|< | - e *sin(n*x)| - cos(n*x)*e
| | || cos(n*x) n*sin(n*x) | |
/ | || - ---------- + ---------- otherwise | |
| || 2 x x 2 x x | |
\ \\ n *e + e n *e + e / /
n2+1e−x(nsin(nx)−cos(nx))
1 cos(n) n*sin(n)
------ - -------- + --------
2 2 2
1 + n e + e*n e + e*n
en2+ensin(n)−en2+ecos(n)+n2+11
=
1 cos(n) n*sin(n)
------ - -------- + --------
2 2 2
1 + n e + e*n e + e*n
en2+ensin(n)−en2+ecos(n)+n2+11
Use the examples entering the upper and lower limits of integration.