Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of (1+x)/x
Integral of (sec(x))^3
Integral of x^2*e^(x^3)
Integral of e^x²
Identical expressions
13sin(3x)*e^(-3x)
13 sinus of (3x) multiply by e to the power of ( minus 3x)
13sin(3x)*e(-3x)
13sin3x*e-3x
13sin(3x)e^(-3x)
13sin(3x)e(-3x)
13sin3xe-3x
13sin3xe^-3x
13sin(3x)*e^(-3x)dx
Similar expressions
13sin(3x)*e^(3x)

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

The solution

You have entered [src]

  1                     
  /                     
 |                      
 |               -3*x   
 |  13*sin(3*x)*E     dx
 |                      
/                       
0

$$\int\limits_{0}^{1} e^{- 3 x} 13 \sin{\left(3 x \right)}\, dx$$

Integral((13*sin(3*x))*E^(-3*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{13 du}{3}$ :
  
  $\int \frac{13 e^{- u} \sin{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int e^{- u} \sin{\left(u \right)}\, du = \frac{13 \int e^{- u} \sin{\left(u \right)}\, du}{3}$
    1. Let $u = - u$ .
      
      Then let $du = - du$ and substitute $du$ :
      
      $\int e^{u} \sin{\left(u \right)}\, du$
      
      Use integration by parts, noting that the integrand eventually repeats itself.
      
      For the integrand $e^{u} \sin{\left(u \right)}$ :
      
      Let $u{\left(u \right)} = \sin{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - \int e^{u} \cos{\left(u \right)}\, du$ .
      
      For the integrand $e^{u} \cos{\left(u \right)}$ :
      
      Let $u{\left(u \right)} = \cos{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)} + \int \left(- e^{u} \sin{\left(u \right)}\right)\, du$ .
      
      Notice that the integrand has repeated itself, so move it to one side:
      
      $2 \int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)}$
      
      Therefore,
      
      $\int e^{u} \sin{\left(u \right)}\, du = \frac{e^{u} \sin{\left(u \right)}}{2} - \frac{e^{u} \cos{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{e^{- u} \sin{\left(u \right)}}{2} - \frac{e^{- u} \cos{\left(u \right)}}{2}$
    So, the result is: $- \frac{13 e^{- u} \sin{\left(u \right)}}{6} - \frac{13 e^{- u} \cos{\left(u \right)}}{6}$
  Now substitute $u$ back in:
  
  $- \frac{13 e^{- 3 x} \sin{\left(3 x \right)}}{6} - \frac{13 e^{- 3 x} \cos{\left(3 x \right)}}{6}$
Method #2
1. Let $u = - 3 x$ .
  
  Then let $du = - 3 dx$ and substitute $\frac{13 du}{3}$ :
  
  $\int \frac{13 e^{u} \sin{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int e^{u} \sin{\left(u \right)}\, du = \frac{13 \int e^{u} \sin{\left(u \right)}\, du}{3}$
    1. Use integration by parts, noting that the integrand eventually repeats itself.
      
      For the integrand $e^{u} \sin{\left(u \right)}$ :
      
      Let $u{\left(u \right)} = \sin{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - \int e^{u} \cos{\left(u \right)}\, du$ .
      
      For the integrand $e^{u} \cos{\left(u \right)}$ :
      
      Let $u{\left(u \right)} = \cos{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)} + \int \left(- e^{u} \sin{\left(u \right)}\right)\, du$ .
      
      Notice that the integrand has repeated itself, so move it to one side:
      
      $2 \int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)}$
      
      Therefore,
      
      $\int e^{u} \sin{\left(u \right)}\, du = \frac{e^{u} \sin{\left(u \right)}}{2} - \frac{e^{u} \cos{\left(u \right)}}{2}$
    So, the result is: $\frac{13 e^{u} \sin{\left(u \right)}}{6} - \frac{13 e^{u} \cos{\left(u \right)}}{6}$
  Now substitute $u$ back in:
  
  $- \frac{13 e^{- 3 x} \sin{\left(3 x \right)}}{6} - \frac{13 e^{- 3 x} \cos{\left(3 x \right)}}{6}$
Now simplify:

$- \frac{13 \sqrt{2} e^{- 3 x} \sin{\left(3 x + \frac{\pi}{4} \right)}}{6}$
Add the constant of integration:

$- \frac{13 \sqrt{2} e^{- 3 x} \sin{\left(3 x + \frac{\pi}{4} \right)}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{13 \sqrt{2} e^{- 3 x} \sin{\left(3 x + \frac{\pi}{4} \right)}}{6}+ \mathrm{constant}$

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        
/

$$\int e^{- 3 x} 13 \sin{\left(3 x \right)}\, dx = C - \frac{13 e^{- 3 x} \sin{\left(3 x \right)}}{6} - \frac{13 e^{- 3 x} \cos{\left(3 x \right)}}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                -3       -3       
13   13*cos(3)*e     13*e  *sin(3)
-- - ------------- - -------------
6          6               6

$$- \frac{13 \sin{\left(3 \right)}}{6 e^{3}} - \frac{13 \cos{\left(3 \right)}}{6 e^{3}} + \frac{13}{6}$$

                -3       -3       
13   13*cos(3)*e     13*e  *sin(3)
-- - ------------- - -------------
6          6               6

$$- \frac{13 \sin{\left(3 \right)}}{6 e^{3}} - \frac{13 \cos{\left(3 \right)}}{6 e^{3}} + \frac{13}{6}$$

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.

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2