How do you integrate #(t^2)e^(4t)#?

Question

Charles Arocha · Accepted Answer

#(t^2e^(4t))/4-1/2((te^(4t))/4-e^(4t)/16)+C#

We have #intt^2e^(4t)dt#. According to Integration by Parts, #intf(t)g(t)dt=f(t)intg(t)dt-intf'(x)(intg(t)dt)dt#

Here, #f(t)=t^2# and #g(t)=e^(4t)#. So we input:

#t^2inte^(4t)dt-int(t^2)'(inte^(4t)dt)dt#

A logical route to take is to find #inte^(4t)dt#

According to Integration by Substitution, #intf(g(t))g'(t)dt=intf(u)du#, where #u=g(t)#. We can write the above as:

#1/4inte^(4t)4dt#

#1/4inte^(u)du#

#1/4e^u#

#e^(4t)/4#. So we input:

#(t^2e^(4t))/4-1/2intte^(4t)dt#

For the integral, use integration by parts:

#tinte^(4t)dt-intt'(inte^(4t)dt)dt#

Since we know that #inte^(4t)dt# is:

#(te^(4t))/4-inte^(4t)/4dt#

#(te^(4t))/4-1/4*1/4e^(4t)#

#(te^(4t))/4-e^(4t)/16#

This can be entered into our earlier computations:

#(t^2e^(4t))/4-1/2((te^(4t))/4-e^(4t)/16)#

Include the integration constant:

#(t^2e^(4t))/4-1/2((te^(4t))/4-e^(4t)/16)+C#

Paisley Johnson · Answer

undefined To integrate (t^2)e^(4t), you can use integration by parts. Let u = t^2 and dv = e^(4t) dt. Then, differentiate u to get du = 2t dt and integrate dv to get v = (1/4)e^(4t). Now, apply the integration by parts formula: ∫ u dv = uv - ∫ v du. Substituting the values, we get: ∫ (t^2)e^(4t) dt = (t^2)(1/4)e^(4t) - ∫ (1/4)e^(4t) * 2t dt. Simplifying, we have: ∫ (t^2)e^(4t) dt = (1/4)t^2e^(4t) - (1/2)∫ te^(4t) dt. Now, we need to integrate te^(4t). Again, we use integration by parts with u = t and dv = e^(4t) dt. Differentiating u, we get du = dt and integrating dv, we obtain v = (1/4)e^(4t). Applying the formula, we have: ∫ te^(4t) dt = (1/4)te^(4t) - (1/4)∫ e^(4t) dt. Simplifying, we get: ∫ te^(4t) dt = (1/4)te^(4t) - (1/16)e^(4t). Finally, substituting this result back into the earlier equation, we get: ∫ (t^2)e^(4t) dt = (1/4)t^2e^(4t) - (1/2)((1/4)te^(4t) - (1/16)e^(4t)) + C, where C is the constant of integration. Simplifying further, we obtain the final result: ∫ (t^2)e^(4t) dt = (1/4)t^2e^(4t) - (1/8)te^(4t) + (1/32)e^(4t) + C.