Evaluate The Integral

Question

First make a substitution and then use integration by parts to evaluate the integral. please show the step by step process. If possible please attach a video going through it. The final answer is 4/e.

Answer 1

Step 1: Get rid of sin(2t) by using the trig identity sin(2x) = 2sin(x)cos(x):

₀∫^πe^cos(t)sin(2t) dt = ₀∫^πe^cos(t)(2sin(t)cos(t)) dt

then move the constant out front:

2₀∫^πe^cos(t)(sin(t)cos(t)) dt then rearrange the sine and cosine to get:

2₀∫^πe^cos(t)cos(t) sin(t)dt

Step 2, let w = cos(t) which makes dw = -sin(t) dt or (-1)dw = sin(t) dt

then substitute to turn the integral into:

2₀∫^πe^w•w•(-1)dw

then move the "-1" constant out front:

-2₀∫^πe^w•w dw

Step 3, Use Integration by Parts

Rule: ∫u dv = uv - ∫v du

Let u = w which makes du = dw

Let dv = e^w which makes v = e^w

So ∫e^w•w dw = w•e^w - ∫e^w dw

Realizing that the anti-derivative of e^w is e^w we get:

∫e^w•w dw = w•e^w - e^w

So -2₀∫^πe^w•w dw = -2(w•e^w - e^w) evaluated for zero to π

Converting back to cos(t), because w = cos(t) we get:

-2(cos(t)•e^cos(t) - e^cos(t)) evaluated for zero to π or, simplifying:

-2e^cos(t)(cos(t) - 1)

Plug in t = π and t = 0 to get:

For t = π:

-2e^cos(π)(cos(π) - 1)

-2e^-1(-1 - 1)

4/e

For t = 0

-2e^cos(0)(cos(0) - 1)

-2e¹(1 - 1)

0

So, putting those together, the integral equals 4/e - 0 or 4/e

Answer 2