Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of e^3
Integral of 4*x*exp(x^2)
Integral of tan^(-1)x
Integral of x*2^x
Identical expressions
cosx*e^x
co sinus of e of x multiply by e to the power of x
cosx*ex
cosxe^x
cosxex
cosx*e^xdx

Solving integrals/ cosx*e^x

Integral of cosx*e^x dx

The solution

You have entered [src]

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

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

Integral(cos(x)*E^x, (x, 0, 1))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

$${{e\,\sin 1+e\,\cos 1}\over{2}}-{{1}\over{2}}$$

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

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

Simplify

Numerical answer [src]

1.37802461354736

1.37802461354736

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of cosx*e^x dx

Limits of integration:

The graph:

Piecewise:

The solution