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Formula sheet

Integration — Formula Sheet

5 formulas from Calculus I, each with what it means, when to use it and the trap it sets.

Reverse power rule

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)

What it says: Add one to the power, then divide by the new power.
When to reach for it: Every polynomial term, and every root or reciprocal rewritten as a power.
Watch out: n = −1 is excluded, because it would divide by zero. That case is ln|x| + C instead.

\int 6 x^{2} d x = 2 x^{3} + C

Differentiate your answer to check — it should give the integrand back.

Standard integrals

\int \frac{1}{x} d x = ln ∣ x ∣ + C, \int e^{x} d x = e^{x} + C, \int cos x d x = sin x + C

What it says: The derivative table, read backwards.
When to reach for it: Whenever the integrand already matches a derivative you know.
Watch out: The absolute value in ln|x| is not decoration — 1/x is defined for negative x too.

\int sin x d x = - cos x + C

The minus sign moves to the other side when the rule is reversed.

Evaluating a definite integral

\int_{a}^{b} f (x) d x = F (b) - F (a)

What it says: Find any antiderivative, then subtract its value at the lower limit from its value at the upper one.
When to reach for it: A definite integral, total accumulated change, or signed area.
Watch out: No +C is needed — it cancels in the subtraction. Area below the axis counts negatively unless the question asks for geometric area.

\int_{0}^{2} 2 x d x = [x^{2}]_{0}^{2} = 4

Upper limit first, then subtract the lower.

Fundamental theorem of calculus

\frac{d}{d x} \int_{a}^{x} f (t) d t = f (x)

What it says: The rate at which accumulated area grows is just the height of the curve at that edge.
When to reach for it: Differentiating an integral with a variable limit.
Watch out: If the upper limit is a function of x, the chain rule applies: the answer is f(g(x))·g′(x).

\frac{d}{d x} \int_{0}^{2 x} e^{t} d t = e^{2 x} \cdot 2

The upper limit 2x contributes its own derivative.

u-substitution

\int f (g (x)) g^{'} (x) d x = \int f (u) d u, u = g (x)

What it says: The chain rule run backwards: name the inside u, and its derivative supplies the du.
When to reach for it: The integrand contains a function and, up to a constant, that function’s derivative.
Watch out: Every x must disappear. If an x survives the substitution, the choice of u was wrong. For a definite integral, convert the limits too or change back before evaluating.

\int 2 x cos (x^{2}) d x = sin (x^{2}) + C

u = x², du = 2x dx — the 2x was already sitting there.

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