State the Evaluation Part of the Fundamental Theorem of Calculus. Then find the definite integrals: (a)`int_2^4 (t^-3)dt` . State the Evaluation Part of the Fundamental Theorem of Calculus. Then...

State the Evaluation Part of the Fundamental Theorem of Calculus. Then find the definite integrals:

(a)`int_2^4 (t^-3)dt` . 

State the Evaluation Part of the Fundamental Theorem of Calculus. Then find the definite integrals:

(a)ʃ(superscript 4)(subscript 2) `int_2^4(t^-3)dt` .   

(b) Find  `int_(-2)^0((e^x) + x)dx` .   

(c) Find`int_2^Ptdt` (the answer is a function of P).  

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lemjay | High School Teacher | (Level 3) Senior Educator

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(a) `int__2^4 t^-3dt`

To start, determine the indefinite integral. To do so, use the power rule of integral which is `int u^n du = u^(n+1)/(n+1) + C` .

`F(t) = int t^-3dt =t^-2/(-2) + C = -t^(-2)/2 + C `

`F(t) = -1/(2t^2) + C`

Then, apply the Fundamental Theorem of calculus to determine the definite integral.

`int_2^4 t^(-3)dt = F(4)-F(2) = (-1/(2*4^2)) - (-1/(2*2^2)) = -1/32 - (-1/8) `

                                  `=-1/32 + 1/8 = -1/32 + 4/32 = 3/32`

Hence, `int_2^4 t^(-3)dt = 3/32` .

(b) `int_(-2)^0 (e^x+x)dx`

Determine the indefinite integral.

`F(x) = int(e^x+x)dx = inte^xdx + intxdx = e^x + x^2/2 + C`

Use the Fundamental Theorem of Calculus.

`int_(-2)^0 (e^x+x)dx = F(0) - F(-2) = (e^0 + 0^2/2) - (e^(-2) + (-2)^2/2 )`

                                          `=(1+0) - (1/e^2 + 4/2) = 1 - 1/e^2 - 2 = -1-1/e^2`

So, `int_(-2)^0 (e^x+x)dx = -1-1/e^2` .

(c) `int_2^P tdt`

Determine the indefinite integral.

`F(t) = int tdt = t^2/2 + C`

Apply the Fundamental Theorem of Calculus to evaluate the definite integral.

`int_2^P tdt = F(P)-F(2) = P^2/2 - 2^2/2 = P^2/2 -4/2 = P^2/2 - 2`

Thus, `int_2^P t dt = P^2/2-2 ` .

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