Mister Exam

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e^3xsen2xdx

What you mean?

Integral of e^3xsen2xdx dx

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The solution

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  1                   
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 |   3                
 |  e *x*sin(2*x)*1 dx
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0                     
$$\int\limits_{0}^{1} e^{3} x \sin{\left(2 x \right)} 1\, dx$$
Integral(E^3*x*sin(2*x)*1, (x, 0, 1))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    1. There are multiple ways to do this integral.

      Method #1

      1. Use integration by parts:

        Let and let .

        Then .

        To find :

        1. There are multiple ways to do this integral.

          Method #1

          1. Let .

            Then let and substitute :

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

              1. The integral of sine is negative cosine:

              So, the result is:

            Now substitute back in:

          Method #2

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

            1. Let .

              Then let and substitute :

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

                1. The integral of is when :

                So, the result is:

              Now substitute back in:

            So, the result is:

        Now evaluate the sub-integral.

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

        1. Let .

          Then let and substitute :

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

            1. The integral of cosine is sine:

            So, the result is:

          Now substitute back in:

        So, the result is:

      Method #2

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

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          1. Let .

            Then let and substitute :

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

              1. The integral of is when :

              So, the result is:

            Now substitute back in:

          Now evaluate the sub-integral.

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

          1. Rewrite the integrand:

          2. Integrate term-by-term:

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

              1. Let .

                Then let and substitute :

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

                  1. The integral of cosine is sine:

                  So, the result is:

                Now substitute back in:

              So, the result is:

            1. The integral of a constant is the constant times the variable of integration:

            The result is:

          So, the result is:

        So, the result is:

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                   
 |                                                    
 |  3                       /sin(2*x)   x*cos(2*x)\  3
 | e *x*sin(2*x)*1 dx = C + |-------- - ----------|*e 
 |                          \   4           2     /   
/                                                     
$$\int e^{3} x \sin{\left(2 x \right)} 1\, dx = C + \left(- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}\right) e^{3}$$
The graph
The answer [src]
/  cos(2)   sin(2)\  3
|- ------ + ------|*e 
\    2        4   /   
$$\left(- \frac{\cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)}}{4}\right) e^{3}$$
=
=
/  cos(2)   sin(2)\  3
|- ------ + ------|*e 
\    2        4   /   
$$\left(- \frac{\cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)}}{4}\right) e^{3}$$
Numerical answer [src]
8.74519808563438
8.74519808563438
The graph
Integral of e^3xsen2xdx dx

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