Please find this integral?

Question

∫ (e^-t)(1+3sin(t)) dt

Answer 1

Use the rules of integration: integral of a sum of two functions equals the sum of their intehrals. Thus, you can write

                 I = ∫ e^-t (1+3sin t) dt =I₁ +3I₂ =  ∫ e^-t  dt + 3∫e^-t  sin t dt

                                                       I₁ = - e^-t  + C₁

For the second integral use integration in parts twice. It gives you      

            I₂ = -e^-t sin t + ∫e^-t cos t dt + C₂= -e^-t (sin t + cos t) + ∫e^-t sin t dt] + C₂=

                                     -e^-t (sin t + cos t)]  - I₂ + C₂

Solve for I₂: 

                                       I₂ = - (1/2) e^-t (sin t + cos t) + C₂/2

Thus, we have

                                I = - e^-t[ 1 + (3/2)(sin t + cos t)]  + C₁ + C₂/2

Now you can use the standard formula

                                           sin t + cos t = √2 sin (t + (π/4)) 

and replace C₁ + C₂/2 by one constant, let say C (because they are arbitrary numbers), and rewrite your integral in the final form

                                       I =  - e^-t [ 1 + (3/2) √2 sin (t + (π/4))]  + C  

which is equaivalent to the one already obtained by Robert.

You can use "integration in parts twice" technique  to calculate the integral such as

                                                             ∫e^-t cos t dt

since the second derivatives of sin and cos functions are the same functions taken with positive (or negative) sign. This idea is very important in calculating more complex integrals such as those with infinite llimits of integration or dependneing on parameters, or calculating Laplace transforms and their applications to differential equations. You are coming up to these topics in your studies.

                                         

                                          

 

 

 

Answer 2

I = ∫ (e^-t) sin(t)) dt

= ∫ -sin(t)) de^-t

= -e^-t sin(t) + ∫ (e^-t) cos(t)) dt

= -e^-t sin(t) + ∫ -cos(t)) de^-t

= -e^-t sin(t) - e^-t cos(t) - I

Solve for I,

I = -(1/2)e^-t (sin(t) + cos(t))

∫ (e^-t)(1+3sin(t)) dt

= -e^-t + 3∫ (e^-t) sin(t)) dt + c

= -e^-t - (3/2)e^-t (sin(t) + cos(t)) + c

= -(1/2)e^-t [3sin(t) + 3cos(t) + 2] + c <==Answer