Integrals

1Integration as an Inverse Process of Differentiation

Integration as inverse of differentiation: if F'=f then the integral of f is F+C as taught. The integral of 2x is x^2+C because d/dx(x^2)=2x. Integral is an undo-derivative, not a fancy sum yet (that is definite).

Forgetting +C on an indefinite is a miss.

Figure. Integration undoes differentiation: the slope of F = x^2 at x = 1/2 is 1, which is the value of f = 2x there. The tangent is drawn through that point; both series start from the origin frame.

How it works

1. Ask whose derivative is fThe undo.
2. Add +CThe family.
3. Check by differentiating backThe inverse-test.

2Methods of Integration

Methods: substitution, and the taught standard forms. For 2x(x^2+1)^3, use u=x^2+1. A method is a rewrite, not a new theory. One substitution is enough to name the method.

A random u that does not tidy du is a miss.

Figure. A method is a rewrite that makes the integrand simpler: substitution, parts, or partial fractions. The destination is the same — a form you can finish.

How it works

1. Name a substitution or a standard formThe method.
2. Rewrite and integrateThe work.
3. Differentiate back if unsureThe check.

3Integrals of Some Particular Functions

Particular functions: the taught list (1/(x^2+a^2), e^x, sin, and the rest as the lesson listed). A particular integral is a remembered undo, not a guess from a poster. Copy the list the class used.

Inventing a form not on the list is a steal.

Figure. Particular integrals are stored rewrites, not slogans. 1/x, e^x and sin x each have one standard antiderivative you recall, then differentiate to check.

How it works

1. Match f to a taught formThe particular.
2. Write the matching FThe integral.
3. Keep +C if indefiniteHonest.

4Integration by Partial Fractions

Partial fractions: split a proper rational as taught, then integrate termwise. (3x+1)/((x+1)(x+2)) becomes A/(x+1)+B/(x+2). Partial is a split-method, not a new derivative.

Integrating numerator and denominator separately is a miss.

How it works

1. Factor the denominator and splitPartial fractions.
2. Integrate each termThe method.
3. Keep it proper (or divide first) as taughtThe gate.

5Integration by Parts

Integration by parts: integral of u dv equals uv minus integral of v du as taught. Pick u that simplifies when differentiated. x e^x is the school case if named. Parts is a product-undo, not substitution.

Picking both u and dv as the hard factor is a miss.

Figure. Integration by parts reverses a product rule: pick u and dv, form u v, then subtract the leftover v du. The leftover is the new integral, not the answer.

How it works

1. Choose u and dvThe split.
2. Form uv minus the leftover integralParts.
3. Keep the ILATE-hint only if the lesson used itHonest extra.

6Definite Integral

Definite integral: from a to b of f is a number, F(b)-F(a) after the next heading. Definite is a from-to, not a +C family. From 0 to 2 of 2x is 4. Limits are part of the write.

Leaving +C on a definite is a miss.

Figure. A definite integral names signed area. Four right-end strips under y = x from 0 to 1 have heights 0.25, 0.50, 0.75, 1 and total area 0.625, above the true area 1/2. The strips are the estimate; the slant is the curve.

How it works

1. Write the limits a to bDefinite.
2. It will become a numberThe kind.
3. Keep +C off this writeA number.

7Fundamental Theorem of Calculus

FTC (school): if F'=f then the definite integral from a to b of f is F(b)-F(a) as taught. The theorem is the evaluate-step. Check F'=f before you subtract.

Using any antiderivative-looking formula without a check is a miss.

Figure. The fundamental theorem says the rate at which area from a to x grows is f(x). The dashed marks are the moving right edge x and the fixed left edge a, not a new curve.

How it works

1. Find F with F'=fThe antiderivative.
2. Form F(b)-F(a)FTC.
3. Refuse +C in the subtractIt cancels.

8Evaluation of Definite Integrals by Substitution

Definite by substitution: change limits with u as taught, or substitute then return to x before using a,b. If u=x^2, x from 0 to 2 becomes u from 0 to 4. Limits must match the new variable.

Changing x but keeping old limits on u is a miss.

Figure. A substitution on a definite integral moves both ends. If u = √x, then x = 1 maps to u = 1 and x = 4 maps to u = 2. Do not keep the old x-limits on the new u-integral.

How it works

1. Change the variable and the limitsThe substitute.
2. Integrate in u from the new a to bThe evaluate.
3. Keep limits honest to uThe caution.

9Some Properties of Definite Integrals

Properties: swapping limits flips the sign; split at c; even/odd if taught. A property is a rewrite of limits or a split, not a new FTC. From 0 to 2 of 2x splits as 0-to-1 plus 1-to-2.

Swapping limits and keeping the same sign is a miss.

Figure. An even integrand gives equal areas on either side of zero. Compute one side and double it; do not add a second independent calculation that can drift.

How it works

1. Name the property (swap, split, even/odd)The rewrite.
2. Apply it before evaluating if it helpsThe use.
3. Keep FTC as the evaluate, this as the tidyTwo tools.