What Is Integration? The Reverse of Derivatives Explained

An integral is the other half of calculus. Derivatives tell you how fast something is changing; integrals tell you how much has accumulated. Once you see that those two ideas are mirror images of each other, a huge chunk of calculus clicks into place.

The antiderivative: undoing a derivative

If d/dx [F(x)] = f(x), then F(x) is called an antiderivative of f(x). The notation for the indefinite integral is ∫ f(x) dx = F(x) + C. The + C matters: since the derivative of any constant is zero, there are infinitely many antiderivatives differing only by a constant.

The power rule in reverse

Differentiation raises the exponent by multiplying and lowering the power: d/dx [xⁿ] = n·xⁿ⁻¹. Integration does the opposite — raise the power by one, then divide by the new exponent: ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C, provided n ≠ −1.

Worked example

Find ∫ 3x² dx. Raise the power: 2+1 = 3. Divide: 3x³ / 3 = x³. Add the constant: answer is x³ + C. Check: d/dx [x³ + C] = 3x². Correct.

The definite integral and area

The definite integral ∫[a to b] f(x) dx gives the signed area between the curve y = f(x) and the x-axis from x = a to x = b. Positive where f is above the axis, negative where f is below. The Fundamental Theorem of Calculus connects the two ideas: ∫[a to b] f(x) dx = F(b) − F(a).

Definite integral example

Find the area under y = 2x from x = 1 to x = 3. Antiderivative: x². Evaluate: F(3) − F(1) = 9 − 1 = 8. The signed area is 8 square units. You can verify geometrically: the region is a trapezoid with parallel sides 2 and 6 and width 2, giving area (2+6)/2 · 2 = 8.

Common integration rules

• ∫ k dx = kx + C (constant rule).
• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1 (power rule).
• ∫ 1/x dx = ln|x| + C (the n = −1 special case).
• ∫ eˣ dx = eˣ + C (exponential).
• ∫ cos x dx = sin x + C and ∫ sin x dx = −cos x + C (trig).

Substitution: the chain rule in reverse

When the integrand contains a composite function, u-substitution often works. Let u = g(x), compute du = g'(x) dx, rewrite the integral in terms of u only, integrate, then substitute back. The key is identifying which part of the integrand plays the role of g'(x).

Integration is one of the topics where seeing each step written out slowly makes the biggest difference. Paste your integral into Solvequill and the explanation video will show the substitution, the evaluation, and the check — all in real time.