Practice Integration
A definite integral measures signed accumulation over an interval, while antiderivatives reverse differentiation.
Distinguish signed area from the total geometric area.
Exercises with solutions
Integration reverses differentiation: increase the exponent by one and divide by the new exponent. For a definite integral, substitute the limits into an antiderivative and take the difference upper minus lower limit.
Exercise 1
Find an antiderivative of f(x) = 2x.
Show solution and working
F(x) = x²
The exponent becomes 2 and you divide by 2: 2 ÷ 2 · x² = x². Check: the derivative of x² is 2x again.
Exercise 2
Find an antiderivative of f(x) = x².
Show solution and working
F(x) = x³/3
The new exponent is 3, so x³ ÷ 3. Check: the derivative of x³/3 is 3x²/3 = x².
Exercise 3
Evaluate the integral of 2x from 0 to 3.
Show solution and working
9
An antiderivative is x². Substitute the limits: 3² − 0² = 9.
Exercise 4
Evaluate the integral of x² from 0 to 3.
Show solution and working
9
An antiderivative is x³/3. Substituting gives 27/3 − 0/3 = 9.
Common mistake
Forgetting the constant of integration C for an indefinite integral, even though there are infinitely many antiderivatives. For a definite integral C is not needed because it cancels in the difference.
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