Question
Use the Fundamental Theorem of Calculus to evaluate the followir 4L 4t5/3e-t/2 dt46 cos(t) dt
Use the Fundamental Theorem of Calculus to evaluate the followir 4L 4t5/3e-t/2 dt 46 cos(t) dt
Answers
Use the Fundamental Theorem of Calculus given in Theorem 5.5 .2 to find the indicated derivative. $$ \frac{d}{d x} \int_{3}^{6 x-1} \sqrt{4 t+9} d t $$
Let's talk about this question. Uh So we have to use the fundamental theorem of calculus or to evaluate is given definite integral. So the definite integral is from a negative 5 to 4. These are the limits and we have to integrity square DT. So the integration of t square is going to be tier is 22 plus 1/2 plus one within the limits of minus four minus 5 to 4. So this becomes T cube over three within the limits of minus +5241 of our three is the constraints that comes outside And we have the TQ remaining which which is in the limits of -5-4. So if you place the parliament we have four q minus. If you place the lower element we have minus five cube. All right. Uh So this comes out as let's continue over here. One of what freedom means as it is for cuba 64 minus minus five cuba's minus 1 25. So we have now won two of 64 minus of minus 1 25 which means that is 1 25 plus 64 which is 1 89 and 1 89 Over three is 63, which is the final answer. Thank you.
For this problem, we are asked to use the fundamental theorem of calculus to find the derivative of the integral from two to T. Of three X squared minus two X to the power of 60 X. So the first thing that we can do is note that by the fundamental theorem of calculus the expression above is going to equal D by D. T. Of the are not integral. Excuse me, D by D T. Of capital F. Of T- Capital F. of two. Where capital F is the anti derivative of our original function. The integrated up there. So taking the derivative of this, we would be just left with D by D. T. Of FFT. Or we can write that as F. Prime of T. But by the definition of the fundamental theorem of calculus F prime of T, it's going to be our original function. So we'll be left with three T squared minus two T. All to the power of six.
In this question we have to integrate the function That is in the limits of zero do 4. And it is where Gadhafi? Okay, first of all we can integrate it first of all or that We have to like it as serious people. one x 2. Now we have to apply the formula of integration. That is access to the power and its integration is access to the power and by endless one divided by endless one. So we got to TVs to the power one by two plus one upon one by two plus one. And we have to put the limits here. First of all I am going to Simplify it. We get here to buy three Here. S to the power three bottles and now I'm going to pull the limits see it? Okay, first off what we've got here before, So we got here two x 3, Order to depart three x 2-. Then we put here zero. Then we got here to buy three in 20 Okay, No sea food Can be written as two square. So we got to square rescue the power three x 2 zero. And simply now we get here To buy three Angel 8 years, okay? And eight multiplied by two. That is 16 by three. So 16 by three is the answer for this given question here. Okay, Thank you.
For this problem, we are asked to use the fundamental theorem of calculus to find the derivative of the integral from 12 X of the function Lawn of T D. T. So the first thing that we know from the fundamental theorem of calculus is that our statement there are equipped or are expression there, I should say, is going to equal D by dx of capital F of X- Capital F. of one. Where you know that F. Prime of X, the derivative of our anti derivative should be Lawn of X. But we can also see that if we take the derivative of capital F. Of X minus capital F. Of one, we're just going to be left with F. Prime of X. So we should be left with Lawn of X as a result.