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Integral of d{x}:
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Integral of e^(−3x)cos2x
Integral of sin^4(3x)cos(3x)
Identical expressions
e^(−3x)cos2x
e to the power of (−3x) co sinus of e of 2x
e(−3x)cos2x
e−3xcos2x
e^−3xcos2x
e^(−3x)cos2xdx

Integral of e^(−3x)cos2x dx

The solution

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 oo                  
  /                  
 |                   
 |   -3*x            
 |  e    *cos(2*x) dx
 |                   
/                    
0

$$\int\limits_{0}^{\infty} e^{- 3 x} \cos{\left(2 x \right)}\, dx$$

Simplify

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{- 3 x} \cos{\left(2 x \right)}$ :
  
  Let $u{\left(x \right)} = \cos{\left(2 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 3 x}$ .
  
  Then $\int e^{- 3 x} \cos{\left(2 x \right)}\, dx = - \int \frac{2 e^{- 3 x} \sin{\left(2 x \right)}}{3}\, dx - \frac{e^{- 3 x} \cos{\left(2 x \right)}}{3}$ .
2. For the integrand $\frac{2 e^{- 3 x} \sin{\left(2 x \right)}}{3}$ :
  
  Let $u{\left(x \right)} = \frac{2 \sin{\left(2 x \right)}}{3}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 3 x}$ .
  
  Then $\int e^{- 3 x} \cos{\left(2 x \right)}\, dx = \int \left(- \frac{4 e^{- 3 x} \cos{\left(2 x \right)}}{9}\right)\, dx + \frac{2 e^{- 3 x} \sin{\left(2 x \right)}}{9} - \frac{e^{- 3 x} \cos{\left(2 x \right)}}{3}$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $\frac{13 \int e^{- 3 x} \cos{\left(2 x \right)}\, dx}{9} = \frac{2 e^{- 3 x} \sin{\left(2 x \right)}}{9} - \frac{e^{- 3 x} \cos{\left(2 x \right)}}{3}$
  
  Therefore,
  
  $\int e^{- 3 x} \cos{\left(2 x \right)}\, dx = \frac{2 e^{- 3 x} \sin{\left(2 x \right)}}{13} - \frac{3 e^{- 3 x} \cos{\left(2 x \right)}}{13}$
Now simplify:

$\frac{\left(2 \sin{\left(2 x \right)} - 3 \cos{\left(2 x \right)}\right) e^{- 3 x}}{13}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                           
 |                                     -3*x      -3*x         
 |  -3*x                   3*cos(2*x)*e       2*e    *sin(2*x)
 | e    *cos(2*x) dx = C - ---------------- + ----------------
 |                                13                 13       
/

$${{e^ {- 3\,x }\,\left(2\,\sin \left(2\,x\right)-3\,\cos \left(2\,x \right)\right)}\over{13}}$$

Simplify

The answer [src]

3/13

$$\frac{3}{13}$$

3/13

$$\frac{3}{13}$$

Simplify

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

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Identical expressions

Integral of e^(−3x)cos2x dx

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The solution