MAT159 starts from the rigorous integral theory built in MAT158: the Fundamental Theorem of Calculus, elementary transcendental functions, and then the practical techniques for evaluating integrals. Finding antiderivatives takes pattern-matching, not a formula, so this unit reviews the foundation before building the standard toolkit.
Unlike differentiation, integration has no mechanical recipe: there is no "product rule for antiderivatives." These techniques are the toolkit built up over centuries of trial, and they reveal that elementary antiderivatives are the exception, not the rule.
The bell curve $e^{-x^2}$ has no elementary antiderivative, so statistics relies on numerical integration of it. Every p-value depends on that fact. Partial fractions decompositions are also exactly how control theory inverts Laplace transforms.
Liouville proved in the 1830s that integrals like $\int e^{-x^2}$ cannot be written in elementary terms; the 1968 Risch algorithm lets computers decide when a closed form exists at all — a rare case of a fully mechanical answer to "can this be done by hand?"
The techniques below only make sense because the FTC connects derivatives to definite integrals. In proof-based calculus, you should be able to state the hypotheses each time, especially continuity of the integrand.
If $f$ is continuous on an interval and $F(x)=\int_a^x f(t)\,dt$, then $F'(x)=f(x)$. If $G'=f$ on $[a,b]$, then $$\int_a^b f(x)\,dx=G(b)-G(a).$$
For $H(x)=\int_1^{x^2}\ln(1+t^2)\,dt$, combine FTC Part 1 with the chain rule: $H'(x)=2x\ln(1+x^4)$.
The official MAT159 topic list includes elementary transcendental functions because they are the main sources and targets of integration techniques. The rules below should be memorized with their domain conditions.
$\dfrac{d}{dx}e^x=e^x$, $\dfrac{d}{dx}\ln x=1/x$ for $x>0$, $\dfrac{d}{dx}\sin x=\cos x$, $\dfrac{d}{dx}\cos x=-\sin x$, and $\dfrac{d}{dx}\arctan x=1/(1+x^2)$. Therefore $\int e^x dx=e^x+C$, $\int dx/x=\ln|x|+C$, and $\int dx/(1+x^2)=\arctan x+C$.
$\int \dfrac{2x}{x^2-1}\,dx=\ln|x^2-1|+C$ on any interval that avoids $x=\pm1$. The absolute value is not decoration: the logarithm formula is local on intervals where the denominator keeps one sign.
$\displaystyle\int f(g(x))g'(x)\,dx=\int f(u)\,du$ (substitution, $u=g(x)$) and $\displaystyle\int u\,dv=uv-\int v\,du$ (parts, from the product rule run backwards).
$\int x e^x\,dx$: let $u=x,\ dv=e^x dx$, so $du=dx,\ v=e^x$.
$\int x^2 e^x\,dx$.
$\int e^x\cos x\,dx$.
Any proper rational function $P(x)/Q(x)$ (degree of $P$ less than degree of $Q$) splits into a sum of simpler pieces, one for each factor of $Q$.
A factor $(x-a)$ contributes $\dfrac{A}{x-a}$; a repeated factor $(x-a)^k$ contributes $\dfrac{A_1}{x-a}+\cdots+\dfrac{A_k}{(x-a)^k}$; an irreducible quadratic $(x^2+bx+c)$ contributes $\dfrac{Ax+B}{x^2+bx+c}$.
$\dfrac{1}{x^2-1}=\dfrac{A}{x-1}+\dfrac{B}{x+1}$.
$\dfrac{x}{(x-1)^2}=\dfrac{A}{x-1}+\dfrac{B}{(x-1)^2}$.
$\dfrac{1}{x(x^2+1)}=\dfrac{A}{x}+\dfrac{Bx+C}{x^2+1}$.
For $\sqrt{a^2-x^2}$ use $x=a\sin\theta$; for $\sqrt{a^2+x^2}$ use $x=a\tan\theta$; for $\sqrt{x^2-a^2}$ use $x=a\sec\theta$. Each turns the radical into a single trig factor via a Pythagorean identity.
$\int\dfrac{dx}{\sqrt{4-x^2}}$, using $x=2\sin\theta$, $dx=2\cos\theta\, d\theta$, $\sqrt{4-x^2}=2\cos\theta$.
$\int\sin^2x\,dx$, using $\sin^2x=\tfrac12(1-\cos2x)$.
For $\int\sin^mx\cos^nx\,dx$: if either power is odd, peel off one factor and substitute with the other function; if both are even, use the half-angle identities repeatedly.
• Convert everything — including $dx$ — to the new variable; do not leave a
mix of $x$ and $u$.
• With definite integrals under substitution, either change the limits or
convert back before evaluating — never mix the two.
• FTC Part 1 differentiates an integral with a variable endpoint; FTC Part 2
evaluates a definite integral using an antiderivative. Keep the two roles
separate.
• $\int dx/x=\ln|x|+C$, not just $\ln x+C$, unless the interval is already
restricted to $x>0$.
• Partial fractions require a proper fraction; divide first if the
numerator's degree is not smaller.
• After a trig substitution, convert back to $x$ using a reference triangle,
including for $\theta$ that appears alone (as $\arcsin$, $\arctan$, etc.).
• Not every elementary function has an elementary antiderivative — do not hunt
forever for a closed form that provably does not exist.
Work each one fully by hand before checking.
$\int x\cos(x^2)\,dx$.
$\int \ln x\,dx$.
$\int_0^1 x e^{-x}\,dx$.
$\int\dfrac{dx}{x(x-1)^2}$.
$\int\dfrac{dx}{x^2+4}$.
$\int\dfrac{dx}{x^2\sqrt{x^2+9}}$.
$\int e^{2x}\sin x\,dx$.
Why can't you find a closed-form antiderivative of $\int\dfrac{\sin x}{x}\,dx$ by these techniques, even though the function is perfectly smooth?
Differentiate $F(x)=\int_0^{\sin x} e^{t^2}\,dt$.
Evaluate $\int \dfrac{3x^2}{1+x^6}\,dx$.
Course coverage. This unit is part of the complete MAT159H1 topic map. It should cover the following items before you move on:
Use the current UofT MAT159H1 calendar topics as the coverage spine, and pair lecture notes with a Spivak-style proof-calculus text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.
The integration spine is: identify the derivative rule being reversed, carry all domain and endpoint conditions, and verify by differentiating the final answer.
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
Differentiate $\int_1^{e^x}\ln(1+t^2)dt$.
Evaluate $\int x^2\ln x\,dx$ for $x>0$.
Integrate $\int dx/(x^2+x)$.
Why should $\int e^{-x^2}dx$ not be attacked forever with substitutions?