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What is the integral of sec³x?

$\int \sec^3(x)\,dx$

Answer $\frac12\sec x\tan x+\frac12\ln\left|\sec x+\tan x\right|+C$

Calculus IIIntegration Verified

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Full solution

How do you differentiate xˣ?

$\frac{d}{dx}\left(x^x\right)$

Answer $x^x(\ln x+1),\quad x>0$

Calculus IDifferentiation Verified

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Why does the harmonic series diverge?

$\sum_{n=1}^{\infty}\frac1n$

AnswerIts partial sums grow without bound; grouping terms gives infinitely many blocks worth at least ½.

Calculus IIInfinite series Verified

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When should integration by parts be used?

$\int u\,dv=uv-\int v\,du$

AnswerUse it when differentiating one factor simplifies it and the other factor can be integrated reliably.

Calculus IIIntegration Verified

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Quick answer

Washer method or shell method?

$V=\pi\int(R^2-r^2)\,dx\quad\text{or}\quad V=2\pi\int rh\,dx$

AnswerChoose the setup that avoids splitting the region and keeps the radius and height easiest to express.

Calculus IIApplications of integration Verified

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