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Formula sheet

Techniques of integration — Formula Sheet

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

Integration by parts

\int u d v = uv - \int v d u

What it says: Trade one integral for another, hoping the new one is easier than the old one.
When to reach for it: A product of two unlike things — a polynomial times a trig, exponential or log.
Watch out: Choose u by LIATE (log, inverse trig, algebraic, trig, exponential): the earlier type becomes u. Picking the wrong u makes the new integral worse, not better.

\int x e^{x} d x = x e^{x} - e^{x} + C

u = x because differentiating it removes the x entirely.

Power-reduction identities

sin^{2} x = \frac{1 - cos 2 x}{2}, cos^{2} x = \frac{1 + cos 2 x}{2}

What it says: An even power of sine or cosine trades itself for a single cosine of double the angle.
When to reach for it: Even powers of sine or cosine, where no substitution presents itself.
Watch out: Only the cosine version carries the plus sign. Getting them the wrong way round flips the answer.

\int sin^{2} x d x = \frac{x}{2} - \frac{sin 2 x}{4} + C

The identity turned an unintegrable square into two standard terms.

Odd trig powers

\int sin^{2 k + 1} x d x : peel one sin x, sin^{2} x = 1 - cos^{2} x

What it says: Split off one factor to become du, and convert everything else to the other function.
When to reach for it: At least one of the two powers is odd.
Watch out: Peel from the odd one. Peeling from the even power leaves you with a square root you cannot substitute.

\int sin^{3} x d x = - cos x + \frac{cos ^{3} x}{3} + C

u = cos x, and the peeled sin x supplied the du.

Partial fractions

\frac{P ( x )}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}

What it says: Split one hard fraction into a sum of easy ones, each integrating to a logarithm.
When to reach for it: A rational function whose denominator factors and whose top has the lower degree.
Watch out: If the numerator degree is not lower, do polynomial division first. A repeated factor needs both A/(x−a) and B/(x−a)².

\frac{1}{x ^{2} - 1} = \frac{1/2}{x - 1} - \frac{1/2}{x + 1}

Each piece integrates to a log, so the answer is a difference of logs.

Improper integral

\int_{a}^{\infty} f (x) d x = b \to \infty lim \int_{a}^{b} f (x) d x

What it says: Replace the infinity with a letter, integrate normally, then take the limit.
When to reach for it: An infinite limit, or an integrand that blows up somewhere on the interval.
Watch out: You must write the limit — an answer that treats ∞ as a number loses the marks even when the value is right. If the limit does not exist, the integral diverges.

\int_{1}^{\infty} \frac{d x}{x ^{2}} = 1, \int_{1}^{\infty} \frac{d x}{x} diverges

The exponent decides it: p > 1 converges, p ≤ 1 does not.

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