Question

In Exercises \(17-20,\) use parts and solve for the unknown integral. $$\int e^{x} \sin x d x$$

Short answer

\(\int e^{x} \sin x d x = \frac{e^x}{2}(sin(x)-cos(x)) + C \) where C is the constant of integration

Step-by-step solution

Identify Function Association

According to the rule of Liouville, a rule of thumb for deciding which function to consider as 'u' and 'v' in the 'Integration by Parts' formula, we choose sin(x) as 'u' and e^x as 'v' because the integral of e^x is easier to obtain.

Apply Integration by Parts

Applying the formula of Integration by Part, integral(u dv) = u*v - integral(v du), we find that integral(e^x sin(x) dx) = sin(x)*e^x - integral(e^x cos(x) dx)

Reapply Integration by Parts

The integral found in step 2, integral(e^x cos(x) dx), needs to be evaluated further using 'Integration by Parts' method. Thus, taking cos(x) as 'u' and e^x as 'v' this time, we get integral(e^x cos(x) dx) = e^x cos(x) - integral(e^x sin(x) dx)

Substitute and Solve

Now we substitute back the value of integral(e^x cos(x) dx) into step 2. It forms a linear equation in integral(e^x sin(x) dx). The solution of this equation is the value of the unknown integral.