Integrals

Learn what an antiderivative is and how to check one: differentiate the candidate and compare the result with $f$. Then see why every antiderivative of $f$ has the form $F(x) + C$.

Learn to read and write the indefinite integral, naming the integrand, variable of integration and constant of integration. Then see why the constant cannot be left off.

Learn the power rule for antiderivatives: raise the exponent by one and divide by the new exponent, which works for every power of $x$ except $x^{-1}$. Then rewrite roots and reciprocals in power form so the same rule integrates them.

Learn to integrate constant multiples, sums and differences, and to handle a full polynomial term by term with a single constant of integration. Then use one known point to pick out the particular antiderivative from the family.

Learn how to integrate exponential functions by dividing out the constant that differentiation produces, covering the natural exponential, a constant multiple in the exponent, and bases other than $e$.

Learn to integrate $\dfrac{1}{x}$, the one power the power rule cannot handle, as $\ln \lvert x \rvert + C$. Then see why the absolute value is needed for the antiderivative to cover every $x \neq 0$, not just the positive half.

Learn to integrate $\sin x$, $\cos x$ and $\sec^2 x$ by reversing the derivatives that produce them, placing the minus sign where differentiation created it. Then integrate combinations of these terms, including integrands written in quotient form.

Learn to estimate the area under a curve by covering the region with rectangles of equal width, taking each height from $f$ at one sample point per strip. Then build a Riemann sum by hand, using left endpoints or right endpoints, for a small number of strips.

Learn how the left and right Riemann sums bound the true area between them when a function rises or falls across a whole interval, and how using more strips narrows that bound towards one number: the definite integral.

Learn how a region below the $x$-axis counts as its area with a minus sign, and how the regions of a graph combine into two different numbers: the net signed area and the total area. Then work out a region's area from a straight-line graph.

Learn to read and write the definite integral, naming the limits of integration, the integrand and the variable of integration. Then see why the value of a definite integral is the net signed area between the graph and the $x$-axis.

Learn the constant-multiple, sum and difference rules for definite integrals, and why they apply only when both integrals are taken over the same interval. Then use additivity to join integrals over adjacent intervals and to recover a missing piece from the whole.

Learn how exchanging the two limits of a definite integral changes its sign, and why an integral over a zero-width interval is zero. Then compare two functions across an interval to compare their integrals, without evaluating either.

Learn why an odd function integrates to zero over an interval centred on $0$, and why an even function's integral over the same interval is twice its integral over the right half. Then apply each shortcut in both directions, and recognise a function with neither symmetry.

Learn how a definite integral with a moving upper limit builds an accumulation function, and how to read where its running total rises and falls straight off the graph. Then meet the theorem that ties the two halves of calculus together: the height of the curve is the rate at which area accumulates.

Learn how accumulating area builds an antiderivative for every continuous function, even when no formula for one can be written down. Then see how the base point decides which antiderivative you get, and how to differentiate an integral by reading its integrand.

Learn how to spot an integral whose upper limit is a function of $x$ rather than $x$ itself, and how to read off the two factors its derivative is built from. Then multiply those factors to differentiate it, without ever evaluating the integral, and find the derivative's value at a particular $x$.

Learn what the second half of the Fundamental Theorem of Calculus claims: integrating a rate across an interval gives the same number as the net change in an antiderivative across it. Then use it to evaluate definite integrals.

Learn why the antiderivative you pick never changes the value of a definite integral. Then meet the evaluation bracket, the notation that records an evaluation between finding the antiderivative and carrying out the subtraction.

Learn to evaluate a definite integral with the Fundamental Theorem of Calculus, integrating the whole integrand once before substituting the limits of integration. Then apply the same process to exponential, trigonometric and reciprocal integrands.

Learn what the average value of a function on an interval is: the single constant height whose rectangle has the same area as the region under the graph. And then compute it.

Learn to evaluate an integral by changing variable: name the inner function of a composite as $u$, rewrite the whole integral in $u$ and $du$, then integrate and back-substitute to return the answer to $x$.

Learn to choose $u$ for yourself rather than being handed it. Then carry a constant through when the integrand offers only a multiple of the derivative, and judge which integrands have that form at all.

Learn to evaluate a definite integral by substitution without ever returning to $x$. Convert both limits of integration to their $u$-values, then integrate in $u$ and apply the evaluation bracket at those limits.

Learn the two substitutions that end in an exponential or a natural logarithm: substitute for the exponent when its derivative is among the factors, and substitute for the denominator when the numerator is its derivative.

Learn how to integrate a product using integration by parts, a formula that follows from the product rule, and how to apply it in both its function and differential notations with the two parts supplied.

Learn to choose $u$ and $dv$ for yourself using the LIATE guideline, then work a standard integration by parts through to the answer. Then integrate $\ln x$, where there is only one factor to split.

Learn how to apply integration by parts to a definite integral, evaluating the boundary term and the remaining integral between the same limits to reach a number.

Learn to apply integration by parts twice, when the transformed integral is again a product. Then handle an exponential multiplied by a sine or cosine, where the original integral reappears and is found by solving for it.

Learn to choose an integration technique from the form of the integrand alone, deciding between linearity and the standard forms, substitution, and integration by parts. Then carry the choice through to an answer, indefinite or definite, including integrals written in a variable other than $x$.

Extend definite integration to intervals that never end. Replace an infinite limit of integration with a limit process, decide whether the result converges, and see why an integral over the whole line must be split into two independent pieces.