Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of e^x/x^2
Integral of (1+x)/x
Integral of cos^4(2x)
Integral of e^(-x)/x
Identical expressions
cos(x)*exp(two *x)
co sinus of e of (x) multiply by exponent of (2 multiply by x)
co sinus of e of (x) multiply by exponent of (two multiply by x)
cos(x)exp(2x)
cosxexp2x
cos(x)*exp(2*x)dx
Similar expressions
cos(x)*exp(2x+7)
cosx*exp(2*x)

Solving integrals/ cos(x)*exp(2*x)

Integral of cos(x)exp(2x) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |          2*x   
 |  cos(x)*e    dx
 |                
/                 
0

$$\int\limits_{0}^{1} e^{2 x} \cos{\left(x \right)}\, dx$$

Integral(cos(x)*exp(2*x), (x, 0, 1))

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{2 x} \cos{\left(x \right)}$ :
  
  Let $u{\left(x \right)} = \cos{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
  
  Then $\int e^{2 x} \cos{\left(x \right)}\, dx = \frac{e^{2 x} \cos{\left(x \right)}}{2} - \int \left(- \frac{e^{2 x} \sin{\left(x \right)}}{2}\right)\, dx$ .
2. For the integrand $- \frac{e^{2 x} \sin{\left(x \right)}}{2}$ :
  
  Let $u{\left(x \right)} = - \frac{\sin{\left(x \right)}}{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
  
  Then $\int e^{2 x} \cos{\left(x \right)}\, dx = \frac{e^{2 x} \sin{\left(x \right)}}{4} + \frac{e^{2 x} \cos{\left(x \right)}}{2} + \int \left(- \frac{e^{2 x} \cos{\left(x \right)}}{4}\right)\, dx$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $\frac{5 \int e^{2 x} \cos{\left(x \right)}\, dx}{4} = \frac{e^{2 x} \sin{\left(x \right)}}{4} + \frac{e^{2 x} \cos{\left(x \right)}}{2}$
  
  Therefore,
  
  $\int e^{2 x} \cos{\left(x \right)}\, dx = \frac{e^{2 x} \sin{\left(x \right)}}{5} + \frac{2 e^{2 x} \cos{\left(x \right)}}{5}$
Now simplify:

$\frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}}{5}$
Add the constant of integration:

$\frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}}{5}+ \mathrm{constant}$

The answer is:

$\frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                
 |                       2*x                    2*x
 |         2*x          e   *sin(x)   2*cos(x)*e   
 | cos(x)*e    dx = C + ----------- + -------------
 |                           5              5      
/

$$\int e^{2 x} \cos{\left(x \right)}\, dx = C + \frac{e^{2 x} \sin{\left(x \right)}}{5} + \frac{2 e^{2 x} \cos{\left(x \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       2                    2
  2   e *sin(1)   2*cos(1)*e 
- - + --------- + -----------
  5       5            5

$$- \frac{2}{5} + \frac{e^{2} \sin{\left(1 \right)}}{5} + \frac{2 e^{2} \cos{\left(1 \right)}}{5}$$

       2                    2
  2   e *sin(1)   2*cos(1)*e 
- - + --------- + -----------
  5       5            5

$$- \frac{2}{5} + \frac{e^{2} \sin{\left(1 \right)}}{5} + \frac{2 e^{2} \cos{\left(1 \right)}}{5}$$

-2/5 + exp(2)*sin(1)/5 + 2*cos(1)*exp(2)/5

Numerical answer [src]

2.4404648818501

2.4404648818501

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of cos(x)exp(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of cos(x)*exp(2*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of cos(x)exp(2x) dx