Online Calculus II Tutoring

Recognize structure before choosing a technique

An indefinite integral asks for all antiderivatives with respect to the variable $x$. Compare $\int 2x\cos(x^2)\,dx$ and $\int x\cos x\,dx$. What feature tells us how to start each one?

Work through the reasoning

In the first integral, the derivative of the inner expression $x^2$ is the factor $2x$ already present. Introduce the substitution variable $u=x^2$, so $du=2x\,dx$. The integral becomes $\int\cos u\,du=\sin u+C$, hence $\sin(x^2)+C$. Here $C$ is an arbitrary constant. In the second integral, use integration by parts: choose $u=x$ and $dv=\cos x\,dx$, so $du=dx$ and $v=\sin x$. Then $\int x\cos x\,dx=x\sin x-\int\sin x\,dx=x\sin x+\cos x+C$. The letters in this second calculation are fresh choices. Differentiate both answers to verify them.

A zero term limit does not guarantee convergence

Let $n$ be a positive integer indexing the terms $a_n=1/n$. These terms tend to zero. Does the harmonic series $1+\tfrac12+\tfrac13+\cdots$ converge?

Convergence means that the partial sums, the totals of the first finitely many terms, approach a finite number. Keep the terms in order and group them after the initial $1$: first $1/2$, then $1/3+1/4$, then the four terms from $1/5$ through $1/8$.

Each term in a group is at least the reciprocal of its last denominator. The first group equals $1/2$; the next is at least $2\cdot(1/4)=1/2$; the four-term group is at least $4\cdot(1/8)=1/2$.

Each next group doubles both the number of terms and the last denominator, so its lower bound remains $1/2$. After $k$ groups, where $k$ is a positive integer, the partial sum is at least $1+k/2$. These sums grow without bound: the harmonic series diverges, even though $a_n$ tends to zero.

A zero term limit is necessary for convergence, but not sufficient. When the term limit is zero, the nth-term divergence test is inconclusive; another argument is needed.

Self-check: With the same positive integer index $n$, let $b_n=1/(2n)$. Do these terms have a convergent infinite sum? Use the harmonic-series example.

Prepare for your lesson

Bring two or three integrals whose first step was unclear. For each, identify the expression inside a composite function and check whether its derivative is present. If you choose integration by parts, write what you will differentiate and what you will integrate before applying the formula.

An original Nablio example, not a university exam question. Your current course outline determines the material to focus on.