Difficulty: Introductory to Intermediate | Prerequisites: Calculus I (differentiation, basic algebra, trigonometric identities)
Elementary integration is the first serious topic in Calculus II and the natural counterpart to differentiation. If Calculus I asked "given a function, what is its rate of change?", integration asks the reverse: "given a rate of change, what was the original function?" This unit also introduces the definite integral, which computes the net area under a curve between two bounds. You will need fluency with algebraic simplification, exponent rules, and basic trig values before any of this lands properly. If those feel shaky, revisit them first.
Integration reverses differentiation. The power rule for integrals, a handful of trig antiderivatives, and one inverse-trig formula handle every problem in this unit. The Fundamental Theorem of Calculus connects the antiderivative to the definite integral, letting you evaluate area under a curve by plugging in bounds.
Antiderivative (indefinite integral)
A function F(x) whose derivative equals f(x). In simple terms, it is the function you get when you "undo" a derivative.
Definite integral
The integral of f(x) evaluated between a lower limit a and an upper limit b, written as the integral from a to b of f(x) dx. It gives the net signed area between the curve and the x-axis over that interval. Think of it as the total accumulation of a quantity between two points.
Fundamental Theorem of Calculus (FTC)
States that if F is an antiderivative of f on [a, b], then the integral from a to b of f(x) dx equals F(b) minus F(a). In simple terms, to evaluate a definite integral you find the antiderivative, plug in the top bound, plug in the bottom bound, and subtract.
Power rule for integration
The integral of x^n dx equals x^(n+1) / (n+1) + C, provided n is not equal to negative 1. Think of it as "raise the exponent by one, then divide by the new exponent." This is the reverse of the power rule for derivatives.
Limits of integration (bounds)
The values a and b that sit at the bottom and top of the integral sign. They define the interval over which you are computing the area.
Constant of integration (C)
The arbitrary constant added to every indefinite integral, because differentiation of a constant gives zero. For definite integrals the constant cancels out, so it is omitted.
Integrand
The function being integrated, the expression between the integral sign and dx. In simple terms, it is the thing you are asked to find the antiderivative of.
The integral of x^n dx = x^(n+1) / (n+1) + C, where n is not equal to negative 1.
Works for negative and fractional exponents too. For example, the square root of x is x^(1/2), so the integral of sqrt(x) dx = x^(3/2) / (3/2) = 2x^(3/2) / 3 + C.
Always rewrite roots and reciprocals as fractional exponents before integrating. This is the single most important algebraic step in elementary integration.
The integral of k * f(x) dx = k * the integral of f(x) dx.
Pull constants out of the integral first. It simplifies the work and reduces errors.
The integral of [f(x) + g(x)] dx = the integral of f(x) dx + the integral of g(x) dx.
Integrate term by term. This means you can break a polynomial into individual powers and handle each one separately.
Many integrands are not given in a form ready for the power rule. You must simplify first.
Expand products: (u + 3)(u - 4) becomes u^2 - u - 12, then integrate term by term.
Divide through: (6t^2 - t) / (3t) becomes 2t - 1/3, which is straightforward to integrate.
Split fractions with a common denominator: (v^6 + 4v^8) / v^5 becomes v + 4v^3.
Rewrite roots: (4 + x^2) / sqrt(x) becomes 4x^(-1/2) + x^(3/2).
The integration itself is usually simple. The algebra beforehand is where most marks are lost.
The integral of sin(x) dx = -cos(x) + C.
The integral of cos(x) dx = sin(x) + C.
These follow directly from the derivatives of sin and cos. You must know the standard trig values at 0, pi/6, pi/4, pi/3, pi/2, and pi to evaluate definite integrals.
The integral of 1 / (1 + x^2) dx = arctan(x) + C.
This comes up whenever the integrand has the form k / (1 + x^2). Pull the constant out, integrate to get arctan, then evaluate at the bounds.
You need to know the standard arctan values: arctan(1/sqrt(3)) = pi/6, arctan(1) = pi/4, arctan(sqrt(3)) = pi/3.
Find the antiderivative F(x).
Compute F(b) - F(a), where a is the lower bound and b is the upper bound.
No constant of integration is needed for definite integrals.
Common pitfall: forgetting to evaluate the antiderivative at BOTH bounds and subtract. Writing the bracket notation F(x) | from a to b helps keep track.
Rule | Formula |
|---|---|
Power rule | integral of x^n dx = x^(n+1) / (n+1) + C, n ≠ -1 |
Constant multiple | integral of k f(x) dx = k integral of f(x) dx |
Sum / difference | integral of [f ± g] dx = integral of f dx ± integral of g dx |
sin integral | integral of sin(x) dx = -cos(x) + C |
cos integral | integral of cos(x) dx = sin(x) + C |
Inverse tangent | integral of 1/(1 + x^2) dx = arctan(x) + C |
FTC evaluation | integral from a to b of f(x) dx = F(b) - F(a) |
Integration computes accumulated quantities from rates. If you know the velocity of a car at every instant, integration gives you the total distance travelled. Engineers use definite integrals to calculate work done by a variable force, the volume of solids of revolution, and the total charge flowing through a circuit over time. In physics, the area under a force-displacement curve is the work performed, and the area under a velocity-time graph is displacement.
Students often forget to add 1 to the exponent before dividing. The power rule is x^(n+1) / (n+1), not x^n / n. Check your answer by differentiating it back.
Leaving sqrt(x) or 1/x^2 in root or fraction form and trying to integrate directly. You must convert to fractional exponents first: sqrt(x) = x^(1/2), 1/x^2 = x^(-2).
Forgetting to evaluate at BOTH bounds. The definite integral is F(b) minus F(a), not just F(b). The lower bound evaluation is where sign errors creep in.
Confusing the sign of trig antiderivatives. The integral of sin(x) is negative cos(x), not positive. Differentiate your answer to confirm.
⚠️ Algebraic simplification before integration is tested heavily. Expect products to expand, fractions to split, and roots to rewrite as fractional exponents.
⚠️ The power rule with fractional and negative exponents appears in nearly every exam. Be fluent with n = 1/2, 3/2, -1/2, and similar.
⚠️ Trig integrals at standard angles (pi/4, pi/3, pi/6) are common. Memorise cos(pi/4) = sqrt(2)/2 and similar.
⚠️ The arctan integral (1/(1 + x^2)) is a favourite exam item. Know the standard arctan values cold.
⚠️ Arithmetic errors in the FTC evaluation step (especially with fractions and negative bounds) cost the most marks. Write out each substitution step.
Q: True or false: the integral of x^3 dx = x^3 / 3 + C.
A: False. It is x^4 / 4 + C. You add 1 to the exponent and divide by the new exponent.
Q: Fill in the blank: the integral of sin(x) dx = ____.
A: -cos(x) + C.
Q: True or false: to integrate sqrt(x), you can apply the power rule directly without rewriting.
A: Technically the power rule works on x^(1/2), but you must first rewrite sqrt(x) as x^(1/2) to see which exponent to use.
Q: The integral from 1 to 3 of 2 dx equals ____.
A: 2(3) - 2(1) = 4. A constant integrates to that constant times x.
Q: True or false: the integral of 5/(1 + x^2) dx = 5 ln(1 + x^2) + C.
A: False. It is 5 arctan(x) + C. The ln form applies to 1/x, not to 1/(1 + x^2).
This connects to substitution (u-substitution) because once elementary integrals feel automatic, substitution extends your toolkit to composite functions. It also connects to integration by parts, which handles products of functions that the sum rule cannot split. The definite integral concept here is the foundation for applications in Calculus II: computing areas between curves, volumes of revolution, arc length, and work.
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