What is an Integral?
An integral finds the accumulated area under a curve or reverses differentiation. It's one of the two foundational operations of calculus, alongside the derivative.
The indefinite integral of xⁿ is ∫xⁿ dx = x^(n+1)/(n+1) + C (for n ≠ −1). A definite integral from a to b gives the net area under the curve between those bounds.
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Step-by-step worked examples
Find ∫x³ dx.
Apply power rule: ∫xⁿ dx = x^(n+1)/(n+1)+C n=3 → x⁴/4 + C
Evaluate ∫₀² x² dx.
∫x² dx = x³/3 + C Evaluate from 0 to 2: (2³/3) − (0³/3) = 8/3 − 0 = 8/3 ≈ 2.67
Find ∫(4x + 3) dx.
Integrate term by term. ∫4x dx = 2x² ∫3 dx = 3x Result: 2x² + 3x + C
Flashcards
Quick quiz
Q1.∫x⁴ dx = ?
Q2.∫₀¹ x dx = ?
Q3.Why do we add +C in indefinite integrals?
Q4.∫(6x²) dx = ?
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Common mistakes
Forgetting the +C in an indefinite integral. — Correct: Always add +C — there are infinitely many antiderivatives differing by a constant.
Using the power rule ∫xⁿdx=x^(n+1)/(n+1) when n=−1. — Correct: That case is special: ∫x⁻¹ dx = ln|x| + C.
Forgetting to divide by the new exponent when integrating. — Correct: Power rule requires dividing by (n+1), not just raising the power.
Not evaluating both bounds in a definite integral. — Correct: Definite integrals require subtracting F(a) from F(b): F(b) − F(a).
FAQ
What is an integral?
An integral is the accumulated area under a curve, or equivalently, the reverse operation of differentiation.
What is the integral power rule formula?
The power rule for integration is ∫xⁿ dx = x^(n+1)/(n+1) + C, valid for any n ≠ −1.
What are examples of integrals?
∫x³ dx = x⁴/4 + C, and the definite integral ∫₀² x² dx = 8/3, representing the area under x² from 0 to 2.
How to calculate an integral?
Apply the power rule to each term, increasing the exponent by 1 and dividing by the new exponent; add +C for indefinite integrals or evaluate the bounds for definite ones.