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Definite Integral

Problems

Antiderivatives and the Fundamental Theorem of Calculus

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Problem : Find f (t)dt .

It follows from the chain rule and the fundamental theorem of calculus that

f (t)dt = 2 f (t)dt f (x)    

Problem : Find all antiderivatives of f (x) = 1/(1 + x) + 2 cos(2x) .

We guess the antiderivative

F(x) = log(1 + x) + sin(2x)    

and check that F'(x) = f (x) . All other antiderivatives must be of the form F(x) + c for some constant c .

Problem : Compute (3x 2 + 7)dx using the fundamental theorem of calculus.

We choose x 3 + 7x as an antiderivative of 3x 2 + 7 . The fundamental theorem of calculus then gives


3x 2 + 7dx = x 3 +7x|-2 4  
  = (43 +7(4)) - ((- 2)3 + 7(- 2))  
  = 114  

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