Integral Calculator (Definite & Indefinite)

Integral Calculator Guide

The Power Rule for Integration

Integration is the reverse of differentiation, and the most fundamental rule is the power rule. To integrate a term of the form a·xⁿ, you increase the power by one and divide by the new power: ∫a·xⁿ dx = a·x^(n+1)/(n+1) + C. The '+ C' is the constant of integration, included for indefinite integrals because any constant differentiates to zero, so the original function could have had any constant term. A worked example: ∫3x² dx — increase the power from 2 to 3, and divide by 3: (3/3)·x³ = x³ + C. Another: ∫6x dx = 6·x²/2 = 3x² + C. The rule works for any power except n = −1 (since dividing by n+1 = 0 is undefined; ∫x⁻¹ dx = ln|x| + C instead, a special case). For a constant term (x⁰), ∫a dx = ax + C. To integrate a sum of terms, integrate each term separately and add them. This calculator applies the power rule to a·xⁿ terms, showing the working. Integration underpins calculating areas, volumes, distances from velocities, and much of physics and engineering — it's one of the two pillars of calculus alongside differentiation.

Definite vs Indefinite Integrals

There are two kinds of integral, and the distinction matters. An indefinite integral (or antiderivative) gives a general function plus the constant of integration C — for example ∫2x dx = x² + C. It represents the family of all functions whose derivative is the integrand. A definite integral has lower and upper limits and gives a specific number — the net signed area between the curve and the x-axis over that interval. You evaluate it using the Fundamental Theorem of Calculus: find the antiderivative F(x), then compute F(upper) − F(lower). The constant C cancels out, so it isn't needed for definite integrals. A worked example: the definite integral of 2x from 0 to 3 is [x²] from 0 to 3 = 3² − 0² = 9, meaning the area under the line y = 2x between x = 0 and x = 3 is 9 square units. If the curve dips below the x-axis, that area counts as negative, so a definite integral gives the net (signed) area — to find total geometric area you'd handle positive and negative parts separately. This calculator computes both: the indefinite antiderivative with its + C, and the definite value between any two limits you enter.

What Integration Represents

Integration has a powerful geometric and physical meaning that makes it far more than abstract algebra. Geometrically, the definite integral of a function between two points equals the area between the curve and the x-axis over that interval — this is why integration is the tool for finding areas of irregular shapes, volumes of revolution, and areas between curves. Physically, integration accumulates quantities: if you integrate velocity over time, you get displacement (distance travelled); integrate acceleration and you get the change in velocity; integrate a rate of flow and you get total volume. This 'accumulation of small pieces' is the essence of integration — it sums infinitely many infinitesimally thin slices. The connection between integration (accumulation/area) and differentiation (rate of change/slope) is captured by the Fundamental Theorem of Calculus, which states they are inverse operations — one of the most important results in all of mathematics. Real-world applications are everywhere: calculating the work done by a varying force, the centre of mass of an object, the total charge from a current, probabilities from distributions in statistics, and areas and volumes in engineering and design. Understanding what an integral represents — accumulated area or total quantity — makes the algebra meaningful rather than mechanical.

Tips and Common Errors

A few habits prevent the most common integration mistakes. Always remember the constant of integration (+ C) for indefinite integrals — forgetting it is the single most frequent error, and it costs marks in exams. For the power rule, the special case n = −1 doesn't follow the standard pattern: ∫1/x dx = ln|x| + C, not x⁰/0. Don't confuse integration with differentiation — integration increases the power and divides (the reverse of differentiation, which decreases the power and multiplies). For definite integrals, substitute the upper limit first, then subtract the value at the lower limit (F(upper) − F(lower)), and watch the signs carefully, especially with negative limits. Keep track of which variable you're integrating with respect to (the 'dx' tells you). For more complex functions beyond simple powers — products, quotients, composite functions, trigonometric and exponential functions — you need other techniques (integration by substitution, by parts, partial fractions) that this calculator's power-rule focus doesn't cover. Always check your answer by differentiating it: if you integrate correctly, differentiating the result should return the original integrand, which is a reliable self-check. For definite integrals, sanity-check the sign — if the function is positive over the interval, the area should be positive. This calculator handles power-rule integration of a·xⁿ terms with full working.