Week 11 Cheatsheet — Integration

The reference card

Everything you need to evaluate a workshop-style integral. Pin it.

Power & constant rules

Integral	Result
$\int kdx$	$kx+c$
$\int x^{n}dx(n\ne -1)$	$\frac{x^{n+1}}{n+1}+c$
$\int (ax+b{)}^{n}dx$	$\frac{(ax+b{)}^{n+1}}{a(n+1)}+c$

Reciprocal & logarithm

Integral	Result
$\int \frac{1}{x}dx$	$\ln |x|+c$
$\int \frac{1}{ax+b}dx$	$\frac{1}{a}\ln |ax+b|+c$
$\int \frac{a}{ax+b}dx$	$\ln |ax+b|+c$

Exponential

Integral	Result
$\int e^{x}dx$	$e^x+c$
$\int e^{ax+b}dx$	$\frac{1}{a}{e}^{ax+b}+c$

Trigonometric

Integral	Result
$\int \cos (x)dx$	$\sin (x)+c$
$\int \cos (ax+b)dx$	$\frac{1}{a}\sin (ax+b)+c$
$\int \sin (x)dx$	$-\cos (x)+c$
$\int \sin (ax+b)dx$	$-\frac{1}{a}\cos (ax+b)+c$
$\int {\sec }^2(x)dx$	$\tan (x)+c$

The pattern: any rule with $(ax+b)$ in the integrand picks up a $\frac{1}{a}$ out front. That is the chain rule running in reverse.

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Definite integrals — the Fundamental Theorem

$${\int }_a^{b}f(x)dx=F(b)-F(a)$$

where $F$ is any antiderivative of $f$. The $+c$ cancels in the subtraction, so drop it.

Procedure

1. Find an antiderivative $F(x)$.
2. Evaluate at the top limit: $F(b)$.
3. Evaluate at the bottom limit: $F(a)$.
4. Subtract: $F(b)-F(a)$.
5. Check signs and units.

Signed area vs. geometric area

A definite integral is signed. If $f(x)<0$ on part of $[a,b]$, that part contributes a negative number. To get the geometric area:

1. Find roots of $f$ on $[a,b]$.
2. Split the integral at each root.
3. Take $|\cdot |$ of any piece that came out negative.
4. Sum the absolute values.

Quick exemplar: ${\int }_0^3(x^2+2x-15)dx=-27$ (signed). Geometric area $=27{\text{units}}^2$.

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Quick worked examples

Type	Example	Result
Power rule	$\int 4x^{3}dx$	$x^4+c$
Sum / linearity	$\int (2\sin x+3e^x)dx$	$-2\cos x+3e^x+c$
Reciprocal	$\int \frac{1}{x}dx$	$\ln |x|+c$
Exponential with $ax+b$	$\int e^{2x}dx$	$\frac{1}{2}{e}^{2x}+c$
Expand first	$\int x(x+1{)}^{2}dx$	$\frac{x^4}{4}+\frac{2x^3}{3}+\frac{x^2}{2}+c$
Definite polynomial	${\int }_1^3(3x^2+x-2)dx$	$26$
Definite exponential	${\int }_1^3{e}^{2t+1}dt$	$\frac{1}{2}(e^7-e^3)\approx 538.27$
Find the constant	${\int }_1^4(3x^2+ax-5)dx=18$	$a=-4$

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Common mistakes

1. Power rule on $1/x$. That gives division by zero. Use $\ln |x|$.
2. Dropping the $\frac{1}{a}$ on $e^{ax+b}$, $\sin (ax+b)$, $(ax+b{)}^n$. The chain rule reverses to a $\frac{1}{a}$ out front.
3. Forgetting $+c$ on indefinite integrals. Pure mark-loser.
4. Top minus bottom. $F(a)-F(b)$ flips the sign. It is $F(b)-F(a)$.
5. Confusing signed area with geometric area when the curve crosses the x-axis.

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Source-PDF typo

The “Basic Integration Rules” handout prints

$$\int kdx=k+c\text{(typo)}$$

The correct rule is $\int kdx=kx+c$. Several other entries in the same handout omit the integration variable in the integrand (e.g. it prints $\int ndx$ where it should be $\int x^{n}dx$). Trust the corrected versions in the table above.

For the why, additional worked examples, and the Fundamental Theorem walked through, see the in-depth note.