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Question

Evaluate the integral 121x-12x2e2xdx using substitution.


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Solution

Step 1: Substitute the appropriate values in the integral.

Given: 121x-12x2e2xdx

Let 2x=tx=t2

Differentiate the above equation we get 2dx=dtdx=dt2

When x=1t=2 and x=2t=4

Now put the value of 2x,x and dxin the given expression.

=241t2-12t24etdt2

Step 2: Simplify the above expression

=242t-2t2etdt2=241t-1t2etdt...(1)

Now let f(t)=1tand f'(t)=-1t2

Now put values of f(t),f'(t) in (1)

=24f(t)+f'(t)etdt

Step 3: Solve the integral

=[et·f(t)]24[et[f(t)+f'(t)]dt=etf(t)]=[et·1t]24=e4·14-e2·12=e2·e2-24

Hence the required answer is e2e2-24


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