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

Calculus IDifferentiation Verified

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Deep dive

Why does the harmonic series diverge?

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

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

Calculus IIInfinite series Verified

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Full solution

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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Guides and explanations

10 useful matches

01Direct answerWhy is the limit of sin x over x equal to 1?A foundational limit whose real proof comes from geometry, not from plugging in zero and hoping.Calculus I8 min 02Method guideHow to evaluate an indeterminate limitA decision process for 0/0 and ∞/∞ that starts with algebra before reaching for a theorem.Calculus I10 min 03Concept explainerWhat continuity at a point really requiresThree conditions, one precise promise: nearby inputs produce nearby outputs without a break at the point.Calculus I7 min 04Method guideHow and when to use the Squeeze TheoremTrap a difficult function between two easier functions that are forced to meet at the same limit.Calculus I8 min 05Concept explainerInfinite limits and vertical asymptotesInfinity describes unbounded behavior, not a number a function eventually reaches.Calculus I8 min 06Direct answerWhat is the derivative of x to the x?The base and exponent both vary, so neither the ordinary power rule nor the simple exponential rule works alone.Calculus I7 min 07Method guideThe Chain Rule as a structure-reading skillDifferentiate the outside function, keep the inside intact, then multiply by the inside rate.Calculus I9 min 08Method guideImplicit differentiation without losing dy/dxDifferentiate an equation whose y-values are not isolated, treating y as a function of x every time it appears.Calculus I9 min 09Method guideWhen logarithmic differentiation is the clean moveUse logarithms to untangle variable exponents and products or quotients with many factors.Calculus I9 min 10Decision guideProduct Rule or Quotient Rule?Choose the derivative rule from the top-level operation—and simplify first when algebra can remove the choice entirely.Calculus I8 min