How to find the integral of f(x) =x^3 - 5 between [ 1, 2]

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To find the integral of f(x) =x^3 - 5 between [ 1, 2], we first find the indefinite integral of f(x) = x^3 - 5

Int [ x^3 - 5]

=> Int [ x^3] - Int [5]

=> x^4 / 4 - 5x + C

Int [ f(x)] = x^4 / 4 - 5x + C

Now for x = 2, the value of Int [ f(x)] is 2^4 / 4 - 5*2 + C

For x = 1, the value of Int [ f(x)] is 1/4 - 5 + C

The required integral is 2^4 / 4 - 5*2 + C - (1/4 - 5 + C)

=> 4 - 10 + C - 1/4 + 5 - C

=> -5/4

The required result is -5/4

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Given the curve f(x) = x^3 - 5

We need to find the definite integral on the interval [1, 2].

Let us intergrate f(x).

Let F(x) = intg f(x)

==> The defininte integral is :

 I = F(2) - F(1).

==> F(x) = intg (x^3 - 5) dx

              = intg x^3 dx - intg 5 dx

               = x^4/4 - 5x + C

==> F(2) = 2^4/4 - 5*2  + C

              = 4 - 10 + c = -6 + c

==> F(1) = 1/4 - 5 + C = -19/4 + C

==> I = -6 + 19/4 = ( -24 + 19) /4 = -5/4

==> Then, the integral is I = -5/4

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