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Solving integrals/ 13sin(3x)*e^(-3x)

Integral of 13sin(3x)*e^(-3x) dx

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 |  13*sin(3*x)*E     dx
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∫01e−3x13sin⁡(3x) dx\int\limits_{0}^{1} e^{- 3 x} 13 \sin{\left(3 x \right)}\, dx 
Integral((13*sin(3*x))*E^(-3*x), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=3xu = 3 x.

      Then let du=3dxdu = 3 dx and substitute 13du3\frac{13 du}{3}:

      ∫13e−usin⁡(u)3 du\int \frac{13 e^{- u} \sin{\left(u \right)}}{3}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫e−usin⁡(u) du=13∫e−usin⁡(u) du3\int e^{- u} \sin{\left(u \right)}\, du = \frac{13 \int e^{- u} \sin{\left(u \right)}\, du}{3}

        1. Let u=−uu = - u.

          Then let du=−dudu = - du and substitute dudu:

          ∫eusin⁡(u) du\int e^{u} \sin{\left(u \right)}\, du

          1. Use integration by parts, noting that the integrand eventually repeats itself.

            1. For the integrand eusin⁡(u)e^{u} \sin{\left(u \right)}:

              Let u(u)=sin⁡(u)u{\left(u \right)} = \sin{\left(u \right)} and let dv⁡(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

              Then ∫eusin⁡(u) du=eusin⁡(u)−∫eucos⁡(u) du\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - \int e^{u} \cos{\left(u \right)}\, du.

            2. For the integrand eucos⁡(u)e^{u} \cos{\left(u \right)}:

              Let u(u)=cos⁡(u)u{\left(u \right)} = \cos{\left(u \right)} and let dv⁡(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

              Then ∫eusin⁡(u) du=eusin⁡(u)−eucos⁡(u)+∫(−eusin⁡(u)) du\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)} + \int \left(- e^{u} \sin{\left(u \right)}\right)\, du.

            3. Notice that the integrand has repeated itself, so move it to one side:

              2∫eusin⁡(u) du=eusin⁡(u)−eucos⁡(u)2 \int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)}

              Therefore,

              ∫eusin⁡(u) du=eusin⁡(u)2−eucos⁡(u)2\int e^{u} \sin{\left(u \right)}\, du = \frac{e^{u} \sin{\left(u \right)}}{2} - \frac{e^{u} \cos{\left(u \right)}}{2}

          Now substitute uu back in:

          −e−usin⁡(u)2−e−ucos⁡(u)2- \frac{e^{- u} \sin{\left(u \right)}}{2} - \frac{e^{- u} \cos{\left(u \right)}}{2}

        So, the result is: −13e−usin⁡(u)6−13e−ucos⁡(u)6- \frac{13 e^{- u} \sin{\left(u \right)}}{6} - \frac{13 e^{- u} \cos{\left(u \right)}}{6}

      Now substitute uu back in:

      −13e−3xsin⁡(3x)6−13e−3xcos⁡(3x)6- \frac{13 e^{- 3 x} \sin{\left(3 x \right)}}{6} - \frac{13 e^{- 3 x} \cos{\left(3 x \right)}}{6}

    Method #2

    1. Let u=−3xu = - 3 x.

      Then let du=−3dxdu = - 3 dx and substitute 13du3\frac{13 du}{3}:

      ∫13eusin⁡(u)3 du\int \frac{13 e^{u} \sin{\left(u \right)}}{3}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫eusin⁡(u) du=13∫eusin⁡(u) du3\int e^{u} \sin{\left(u \right)}\, du = \frac{13 \int e^{u} \sin{\left(u \right)}\, du}{3}

        1. Use integration by parts, noting that the integrand eventually repeats itself.

          1. For the integrand eusin⁡(u)e^{u} \sin{\left(u \right)}:

            Let u(u)=sin⁡(u)u{\left(u \right)} = \sin{\left(u \right)} and let dv⁡(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

            Then ∫eusin⁡(u) du=eusin⁡(u)−∫eucos⁡(u) du\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - \int e^{u} \cos{\left(u \right)}\, du.

          2. For the integrand eucos⁡(u)e^{u} \cos{\left(u \right)}:

            Let u(u)=cos⁡(u)u{\left(u \right)} = \cos{\left(u \right)} and let dv⁡(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

            Then ∫eusin⁡(u) du=eusin⁡(u)−eucos⁡(u)+∫(−eusin⁡(u)) du\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)} + \int \left(- e^{u} \sin{\left(u \right)}\right)\, du.

          3. Notice that the integrand has repeated itself, so move it to one side:

            2∫eusin⁡(u) du=eusin⁡(u)−eucos⁡(u)2 \int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)}

            Therefore,

            ∫eusin⁡(u) du=eusin⁡(u)2−eucos⁡(u)2\int e^{u} \sin{\left(u \right)}\, du = \frac{e^{u} \sin{\left(u \right)}}{2} - \frac{e^{u} \cos{\left(u \right)}}{2}

        So, the result is: 13eusin⁡(u)6−13eucos⁡(u)6\frac{13 e^{u} \sin{\left(u \right)}}{6} - \frac{13 e^{u} \cos{\left(u \right)}}{6}

      Now substitute uu back in:

      −13e−3xsin⁡(3x)6−13e−3xcos⁡(3x)6- \frac{13 e^{- 3 x} \sin{\left(3 x \right)}}{6} - \frac{13 e^{- 3 x} \cos{\left(3 x \right)}}{6}

  2. Now simplify:

    −132e−3xsin⁡(3x+π4)6- \frac{13 \sqrt{2} e^{- 3 x} \sin{\left(3 x + \frac{\pi}{4} \right)}}{6}

  3. Add the constant of integration:

    −132e−3xsin⁡(3x+π4)6+constant- \frac{13 \sqrt{2} e^{- 3 x} \sin{\left(3 x + \frac{\pi}{4} \right)}}{6}+ \mathrm{constant}

The answer is:

−132e−3xsin⁡(3x+π4)6+constant- \frac{13 \sqrt{2} e^{- 3 x} \sin{\left(3 x + \frac{\pi}{4} \right)}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                
 |                                         -3*x       -3*x         
 |              -3*x          13*cos(3*x)*e       13*e    *sin(3*x)
 | 13*sin(3*x)*E     dx = C - ----------------- - -----------------
 |                                    6                   6        
/
∫e−3x13sin⁡(3x) dx=C−13e−3xsin⁡(3x)6−13e−3xcos⁡(3x)6\int e^{- 3 x} 13 \sin{\left(3 x \right)}\, dx = C - \frac{13 e^{- 3 x} \sin{\left(3 x \right)}}{6} - \frac{13 e^{- 3 x} \cos{\left(3 x \right)}}{6}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
                -3       -3       
13   13*cos(3)*e     13*e  *sin(3)
-- - ------------- - -------------
6          6               6
−13sin⁡(3)6e3−13cos⁡(3)6e3+136- \frac{13 \sin{\left(3 \right)}}{6 e^{3}} - \frac{13 \cos{\left(3 \right)}}{6 e^{3}} + \frac{13}{6} 
=
= 
                -3       -3       
13   13*cos(3)*e     13*e  *sin(3)
-- - ------------- - -------------
6          6               6
−13sin⁡(3)6e3−13cos⁡(3)6e3+136- \frac{13 \sin{\left(3 \right)}}{6 e^{3}} - \frac{13 \cos{\left(3 \right)}}{6 e^{3}} + \frac{13}{6} 
13/6 - 13*cos(3)*exp(-3)/6 - 13*exp(-3)*sin(3)/6
Numerical answer [src] 
2.25823622401557
2.25823622401557

    Use the examples entering the upper and lower limits of integration.

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