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Integral of (5sin2x+14e^(3x)) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                          
  /                          
 |                           
 |  /                 3*x\   
 |  \5*sin(2*x) + 14*E   / dx
 |                           
/                            
0                            
$$\int\limits_{0}^{1} \left(14 e^{3 x} + 5 \sin{\left(2 x \right)}\right)\, dx$$
Integral(5*sin(2*x) + 14*E^(3*x), (x, 0, 1))
Detail solution
  1. 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 the exponential function is itself.

          So, the result is:

        Now substitute back in:

      So, the result is:

    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. 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. 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 is when :

                So, the result is:

              Now substitute back in:

            Method #2

            1. Let .

              Then let and substitute :

              1. The integral of is when :

              Now substitute back in:

          So, the result is:

      So, the result is:

    The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                    
 |                                                  3*x
 | /                 3*x\          5*cos(2*x)   14*e   
 | \5*sin(2*x) + 14*E   / dx = C - ---------- + -------
 |                                     2           3   
/                                                      
$$\int \left(14 e^{3 x} + 5 \sin{\left(2 x \right)}\right)\, dx = C + \frac{14 e^{3 x}}{3} - \frac{5 \cos{\left(2 x \right)}}{2}$$
The graph
The answer [src]
                      3
  13   5*cos(2)   14*e 
- -- - -------- + -----
  6       2         3  
$$- \frac{13}{6} - \frac{5 \cos{\left(2 \right)}}{2} + \frac{14 e^{3}}{3}$$
=
=
                      3
  13   5*cos(2)   14*e 
- -- - -------- + -----
  6       2         3  
$$- \frac{13}{6} - \frac{5 \cos{\left(2 \right)}}{2} + \frac{14 e^{3}}{3}$$
-13/6 - 5*cos(2)/2 + 14*exp(3)/3
Numerical answer [src]
92.6062060662436
92.6062060662436

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