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Solving integrals/ exp(7x)*sin(x)

Integral of exp(7x)*sin(x) dx

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The solution

You have entered [src] 
 oo               
  /               
 |                
 |   7*x          
 |  e   *sin(x) dx
 |                
/                 
0
∫0∞e7xsin⁡(x) dx\int\limits_{0}^{\infty} e^{7 x} \sin{\left(x \right)}\, dx 
Integral(exp(7*x)*sin(x), (x, 0, oo))
Detail solution

  1. Use integration by parts, noting that the integrand eventually repeats itself.

    1. For the integrand e7xsin⁡(x)e^{7 x} \sin{\left(x \right)}:

      Let u(x)=sin⁡(x)u{\left(x \right)} = \sin{\left(x \right)} and let dv⁡(x)=e7x\operatorname{dv}{\left(x \right)} = e^{7 x}.

      Then ∫e7xsin⁡(x) dx=e7xsin⁡(x)7−∫e7xcos⁡(x)7 dx\int e^{7 x} \sin{\left(x \right)}\, dx = \frac{e^{7 x} \sin{\left(x \right)}}{7} - \int \frac{e^{7 x} \cos{\left(x \right)}}{7}\, dx.

    2. For the integrand e7xcos⁡(x)7\frac{e^{7 x} \cos{\left(x \right)}}{7}:

      Let u(x)=cos⁡(x)7u{\left(x \right)} = \frac{\cos{\left(x \right)}}{7} and let dv⁡(x)=e7x\operatorname{dv}{\left(x \right)} = e^{7 x}.

      Then ∫e7xsin⁡(x) dx=e7xsin⁡(x)7−e7xcos⁡(x)49+∫(−e7xsin⁡(x)49) dx\int e^{7 x} \sin{\left(x \right)}\, dx = \frac{e^{7 x} \sin{\left(x \right)}}{7} - \frac{e^{7 x} \cos{\left(x \right)}}{49} + \int \left(- \frac{e^{7 x} \sin{\left(x \right)}}{49}\right)\, dx.

    3. Notice that the integrand has repeated itself, so move it to one side:

      50∫e7xsin⁡(x) dx49=e7xsin⁡(x)7−e7xcos⁡(x)49\frac{50 \int e^{7 x} \sin{\left(x \right)}\, dx}{49} = \frac{e^{7 x} \sin{\left(x \right)}}{7} - \frac{e^{7 x} \cos{\left(x \right)}}{49}

      Therefore,

      ∫e7xsin⁡(x) dx=7e7xsin⁡(x)50−e7xcos⁡(x)50\int e^{7 x} \sin{\left(x \right)}\, dx = \frac{7 e^{7 x} \sin{\left(x \right)}}{50} - \frac{e^{7 x} \cos{\left(x \right)}}{50}

  2. Now simplify:

    (7sin⁡(x)−cos⁡(x))e7x50\frac{\left(7 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{7 x}}{50}

  3. Add the constant of integration:

    (7sin⁡(x)−cos⁡(x))e7x50+constant\frac{\left(7 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{7 x}}{50}+ \mathrm{constant}

The answer is:

(7sin⁡(x)−cos⁡(x))e7x50+constant\frac{\left(7 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{7 x}}{50}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                
 |                              7*x      7*x       
 |  7*x                 cos(x)*e      7*e   *sin(x)
 | e   *sin(x) dx = C - ----------- + -------------
 |                           50             50     
/
∫e7xsin⁡(x) dx=C+7e7xsin⁡(x)50−e7xcos⁡(x)50\int e^{7 x} \sin{\left(x \right)}\, dx = C + \frac{7 e^{7 x} \sin{\left(x \right)}}{50} - \frac{e^{7 x} \cos{\left(x \right)}}{50}
Simplify
The answer [src] 
<-oo, oo>
⟨−∞,∞⟩\left\langle -\infty, \infty\right\rangle 
=
= 
<-oo, oo>
⟨−∞,∞⟩\left\langle -\infty, \infty\right\rangle 
AccumBounds(-oo, oo)

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

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