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Integral of d{x}:
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Integral of x*ln(x+1)dx
Identical expressions
e^(5x)*cos(7x)
e to the power of (5x) multiply by co sinus of e of (7x)
e(5x)*cos(7x)
e5x*cos7x
e^(5x)cos(7x)
e(5x)cos(7x)
e5xcos7x
e^5xcos7x
e^(5x)*cos(7x)dx

Solving integrals/ e^(5x)*cos(7x)

Integral of e^(5x)*cos(7x) dx

The solution

You have entered [src]

  1                 
  /                 
 |                  
 |   5*x            
 |  e   *cos(7*x) dx
 |                  
/                   
0

$$\int\limits_{0}^{1} e^{5 x} \cos{\left(7 x \right)}\, dx$$

Simplify

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{5 x} \cos{\left(7 x \right)}$ :
  
  Let $u{\left(x \right)} = \cos{\left(7 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
  
  Then $\int e^{5 x} \cos{\left(7 x \right)}\, dx = \frac{e^{5 x} \cos{\left(7 x \right)}}{5} - \int \left(- \frac{7 e^{5 x} \sin{\left(7 x \right)}}{5}\right)\, dx$ .
2. For the integrand $- \frac{7 e^{5 x} \sin{\left(7 x \right)}}{5}$ :
  
  Let $u{\left(x \right)} = - \frac{7 \sin{\left(7 x \right)}}{5}$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
  
  Then $\int e^{5 x} \cos{\left(7 x \right)}\, dx = \frac{7 e^{5 x} \sin{\left(7 x \right)}}{25} + \frac{e^{5 x} \cos{\left(7 x \right)}}{5} + \int \left(- \frac{49 e^{5 x} \cos{\left(7 x \right)}}{25}\right)\, dx$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $\frac{74 \int e^{5 x} \cos{\left(7 x \right)}\, dx}{25} = \frac{7 e^{5 x} \sin{\left(7 x \right)}}{25} + \frac{e^{5 x} \cos{\left(7 x \right)}}{5}$
  
  Therefore,
  
  $\int e^{5 x} \cos{\left(7 x \right)}\, dx = \frac{7 e^{5 x} \sin{\left(7 x \right)}}{74} + \frac{5 e^{5 x} \cos{\left(7 x \right)}}{74}$
Now simplify:

$\frac{\left(7 \sin{\left(7 x \right)} + 5 \cos{\left(7 x \right)}\right) e^{5 x}}{74}$
Add the constant of integration:

$\frac{\left(7 \sin{\left(7 x \right)} + 5 \cos{\left(7 x \right)}\right) e^{5 x}}{74}+ \mathrm{constant}$

The answer is:

$\frac{\left(7 \sin{\left(7 x \right)} + 5 \cos{\left(7 x \right)}\right) e^{5 x}}{74}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                        
 |                                    5*x      5*x         
 |  5*x                   5*cos(7*x)*e      7*e   *sin(7*x)
 | e   *cos(7*x) dx = C + --------------- + ---------------
 |                               74                74      
/

$${{e^{5\,x}\,\left(7\,\sin \left(7\,x\right)+5\,\cos \left(7\,x \right)\right)}\over{74}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                 5      5       
  5    5*cos(7)*e    7*e *sin(7)
- -- + ----------- + -----------
  74        74            74

$${{7\,e^5\,\sin 7+5\,e^5\,\cos 7}\over{74}}-{{5}\over{74}}$$

                 5      5       
  5    5*cos(7)*e    7*e *sin(7)
- -- + ----------- + -----------
  74        74            74

$$- \frac{5}{74} + \frac{5 e^{5} \cos{\left(7 \right)}}{74} + \frac{7 e^{5} \sin{\left(7 \right)}}{74}$$

Simplify

Numerical answer [src]

16.7159901666974

16.7159901666974

The graph

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

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Identical expressions

Integral of e^(5x)*cos(7x) dx

Limits of integration:

The graph:

Piecewise:

The solution