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Exercises on integrals indefinite and definite

Calculate indefinite and definite integrals using the power rule and Barrow's rule. Integral theory and step-by-step solved exercises, with practice instantly corrected by AI.

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Integrals: quick theory

Integration is the inverse operation of differentiation. The indefinite integral ∫f(x) dx is the set of functions whose derivative is f(x); that is why the constant +C is always added.

Power rule: ∫xn dx = xn+1/(n+1) + C (with n ≠ −1).

The definite integral ∫ between a and b is calculated using Barrow's rule: an antiderivative F is found and F(b) − F(a) is evaluated. It is used to calculate areas.

Step-by-step solved exercises on integrals

Solved examples to show you the method. Then, practice with instant correction.

Exercise 1Calculate ∫ (3x² + 2x) dx.

Step-by-step solution

  1. We integrate term by term using the power rule.
  2. ∫3x² dx = 3·x³/3 = x³; ∫2x dx = 2·x²/2 = x².
  3. We add the constant +C.
Result: ∫(3x² + 2x) dx = x³ + x² + C.

Exercise 2Calculate ∫ (x⁴ − 4x) dx.

Step-by-step solution

  1. ∫x⁴ dx = x⁵/5; ∫−4x dx = −4·x²/2 = −2x².
  2. We add +C.
Result: ∫(x⁴ − 4x) dx = x⁵/5 − 2x² + C.

Exercise 3Calculate the definite integral ∫ from 0 to 2 of 2x dx.

Step-by-step solution

  1. An antiderivative of 2x is x².
  2. Barrow's rule: F(2) − F(0) = 2² − 0² = 4 − 0.
Result: ∫₀² 2x dx = 4.

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This topic is studied in:

2nd year of Bachillerato

Frequently asked questions about integrals

Why is +C added in indefinite integrals?

Because many functions have the same derivative (they differ by a constant). The constant +C represents all those possible antiderivatives.

How do you calculate a definite integral?

Using Barrow's rule: find an antiderivative F of the function and calculate F(b) − F(a), where a and b are the limits of integration. The result is a number (for example, an area).

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