Integral Calculator (Definite & Indefinite)

Integral Calculator Guide

What should I know about 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.

What's the difference between Definite and 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.

What do I need to know about 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.

Worked Example: Evaluating a Definite Integral Step by Step

Consider the definite integral of (2x + 3) with respect to x, evaluated from x = 1 to x = 4. The first step is finding the antiderivative using the power rule on each term: the antiderivative of 2x is x², and the antiderivative of the constant 3 is 3x. So the antiderivative (indefinite integral) is x² + 3x + C. For a definite integral, the constant C cancels out, so it is dropped, and the antiderivative is evaluated at the upper limit, then at the lower limit, with the second result subtracted from the first. At x = 4: (4)² + 3(4) = 16 + 12 = 28. At x = 1: (1)² + 3(1) = 1 + 3 = 4. Subtracting gives 28 − 4 = 24. So ∫₁⁴ (2x + 3) dx = 24. Geometrically, this is the exact area between the line y = 2x + 3 and the x-axis, between x = 1 and x = 4 — which can be double-checked with the trapezoid area formula, since the region is a trapezoid with parallel sides of height 5 (at x=1) and 11 (at x=4) and width 3: area = ½(5+11)(3) = 24, matching the integral exactly.

Common Integration Rules at a Glance

Most integrals a student meets before calculus-based physics or statistics reduce to a short list of standard rules. Memorising this table, and recognising which rule applies to a given expression, is the fastest path to solving integrals without a computer.

For sums of terms, integration is linear: integrate each term separately and add the results, which is exactly what was done in the worked example above for 2x + 3.

Why the Constant of Integration Matters

Every indefinite integral technically has infinitely many correct answers, differing only by a constant, because the derivative of any constant is zero — so adding +5, +100, or +C to an antiderivative never changes its derivative. Writing '+C' is a reminder that an indefinite integral describes a whole family of functions sharing the same slope at every point, not one single curve, and the specific value of C is only pinned down when an extra condition (like a known starting value) is given.

Worked Example: Finding the Area Between Two Curves

To find the area trapped between two curves, integrate the difference between the upper and lower function across the interval where they bound a region. Take f(x) = x and g(x) = x^2 - 2, which intersect where x = x^2 - 2, i.e. x^2 - x - 2 = 0, giving x = -1 and x = 2. On this interval, f(x) is the upper curve, so the area is the integral from -1 to 2 of (x - (x^2 - 2)) dx = integral of (-x^2 + x + 2) dx. The antiderivative is -x^3/3 + x^2/2 + 2x. Evaluating at x=2 gives -8/3 + 2 + 4 = 4.667, and at x=-1 gives 1/3 + 0.5 - 2 = -1.167. Subtracting: 4.667 - (-1.167) = 5.833. The enclosed area is therefore 5.833 square units. Always sketch or at least mentally check which curve sits on top across the whole interval before integrating, since integrating (lower minus upper) by mistake simply flips the sign of the answer rather than producing an error.