Integral of e^7xcos4xdx dx

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  n                 
  -                 
  4                 
  /                 
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 |   7              
 |  E *x*cos(4*x) dx
 |                  
/                   
0

\int\limits_{0}^{\frac{n}{4}} e^{7} x \cos{\left(4 x \right)}\, dx

Integral((E^7*x)*cos(4*x), (x, 0, n/4))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x e^{7}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(4 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = e^{7}$ .

To find $v{\left(x \right)}$ :
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{\cos{\left(u \right)}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{4}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(4 x \right)}}{4}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{e^{7} \sin{\left(4 x \right)}}{4}\, dx = \frac{e^{7} \int \sin{\left(4 x \right)}\, dx}{4}$
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{\sin{\left(u \right)}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(4 x \right)}}{4}$
So, the result is: $- \frac{e^{7} \cos{\left(4 x \right)}}{16}$
Now simplify:

$\frac{\left(4 x \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{7}}{16}$
Add the constant of integration:

$\frac{\left(4 x \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{7}}{16}+ \mathrm{constant}$

The answer is:

\frac{\left(4 x \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{7}}{16}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                                  
 |                                  7      7         
 |  7                     cos(4*x)*e    x*e *sin(4*x)
 | E *x*cos(4*x) dx = C + ----------- + -------------
 |                             16             4      
/

\int e^{7} x \cos{\left(4 x \right)}\, dx = C + \frac{x e^{7} \sin{\left(4 x \right)}}{4} + \frac{e^{7} \cos{\left(4 x \right)}}{16}

Simplify

The answer [src]

   7                         
  e    /cos(n)   n*sin(n)\  7
- -- + |------ + --------|*e 
  16   \  16        16   /

\left(\frac{n \sin{\left(n \right)}}{16} + \frac{\cos{\left(n \right)}}{16}\right) e^{7} - \frac{e^{7}}{16}

=

   7                         
  e    /cos(n)   n*sin(n)\  7
- -- + |------ + --------|*e 
  16   \  16        16   /

\left(\frac{n \sin{\left(n \right)}}{16} + \frac{\cos{\left(n \right)}}{16}\right) e^{7} - \frac{e^{7}}{16}

-exp(7)/16 + (cos(n)/16 + n*sin(n)/16)*exp(7)

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Integral of e^7xcos4xdx dx

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