Integrate ((7e^(x/2))/4) with respect to x within the bounds of x=0 and x=2. (Basic introduction to definite integration)

Check to see if anything can be removed from withing the integral and taken out the front. In this case, 7/4 can be taken out the front (as it is not dependent on x), leaving only the term e^(x/2) to be integrated.Since the derivative of the exponential is still the exponential we can work backwards here (i.e. what differentiates to give e^(x/2)?)Differentiating e^(x/2), by the laws of differentiating the exponential, will give us the differential of the power multiplied by our original function. In this case the differential of the power is 1/2 and the original function is e^(x/2). So we have (e^(x/2))/2The function we are looking to achieve (the one we have inside our current integral; e^(x/2)) is very close to this result found in step 3. All we have to do is multiply by 2 to get the result we want, so that is what we will do.Using the information realised in step 4 we shall try differentiating (2)(e^(x/2)). After calculation, this gives us our desired result of e^(x/2), implying that when you integrate e^(x/2) you get (2)(e^(x/2)).Now we have completed the integration part of the question we are left with: (7/4)[2(e^(x/2))]²₀ Now we must calculate the final value using the bounds given in the question. We want to sub these into our result according to the rules of definite integration.Completing step 7 will give: (7/4) [(2)(e^(2/2)) - (2)(e^(0/2))] = (7/4) [(2)(e^1) - (2)(e^0)] = (7/4)(2e - 2) = (7/4)(2)(e-1)
Simplifying, we now have our final result of (7/2)(e-1).