Calculus Essentials
Evaluate ∫ln(x) dx using integration by parts.
If the curves are x = f(y) and x = g(y), the area is ∫_c^d |f(y) − g(y)| dy, where c and d are the y-values of intersection. This approach is useful when the region is more naturally described horizontally.
LIATE is a guideline for choosing u in integration by parts. In order of priority: L I A T rigonometric, E
Use shells when the axis of revolution is perpendicular to the axis of the variable of integration. For example, when revolving y = f(x) around the y-axis, shells (integrating dx) are often easier than disks (which would require solving for x in terms of y).
Let u = ln(x), dv = dx. Then du = (1/x)dx, v = x. So ∫ln(x) dx = x·ln(x) − ∫x·(1/x) dx = x·ln(x) − x + C = x(ln(x) − 1) + C .
Calculus Essentials
If F(x) = ∫₀ˣ cos(t²) dt, what is F′(x)?
The derivative of eˣ is eˣ . The exponential function is its own derivative, which is one of the key properties that makes e the natural base for exponential functions in calculus.
By the Second Fundamental Theorem of Calculus, F′(x) = cos(x²) . We simply substitute the upper limit variable into the integrand. No antiderivative of cos(t²) is needed.
A Maclaurin series is a Taylor series centered at a = 0 : ∑(n=0 to ∞) f⁽ⁿ⁾(0)/n! · xⁿ. Common examples include the series for eˣ, sin(x), and cos(x). It is simply the special case of a Taylor series at the origin.
The function may have a corner, cusp, vertical tangent, or discontinuity at that point. In each case, the limit defining the derivative fails to exist, so no unique tangent line can be drawn.
Calculus Essentials
What is a Maclaurin series?
The limit of a function f(x) as x approaches a value c
A Maclaurin series is a Taylor series centered at a = 0 : ∑(n=0 to ∞) f⁽ⁿ⁾(0)/n! · xⁿ. Common examples include the series for eˣ, sin(x), and cos(x). It is simply the special case of a Taylor series at the origin.
If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f′(c) = [f(b) − f(a)] / (b − a) . Geometrically, there is a point where the tangent line is parallel to the secant line through the endpoints.
Divide every term in the numerator and denominator by the highest power of x in the denominator. Terms with x in the denominator approach 0, leaving only the leading coefficients. If the degrees are equal, the limit is the ratio of leading coefficients.
Calculus Essentials
When do you need to apply integration by parts twice?
If f(x) = xⁿ for any real number n, then f′(x) = nxⁿ⁻¹ . For example, the derivative of x⁵ is 5x⁴. This rule applies to integer, fractional, and negative exponents.
sin(x) = ∑(n=0 to ∞) (−1)ⁿ x²ⁿ⁺¹/(2n+1)! = x − x³/3! + x⁵/5! − ⋯ . This series converges for all x and contains only odd powers of x, reflecting that sine is an odd function.
When using substitution on a definite integral, you must change the limits of integration to match the new variable u. If u = g(x), the new limits are u(a) and u(b). This eliminates the need to substitute back to x.
Sometimes the resulting integral after one application of parts still requires parts again, such as ∫x²eˣ dx. In some cases, like ∫eˣsin(x) dx, applying parts twice produces the original integral, and you can solve algebraically for it.
Calculus Essentials
What is the shell method for volumes of revolution?
Yes. Differentiability at a point implies continuity at that point. However, the converse is not true — a function can be continuous at a point without being differentiable there (e.g., f(x) = |x| at x = 0).
The shell method uses cylindrical shells: V = 2π∫ₐᵇ x·f(x) dx when revolving around the y-axis. Each shell has radius x, height f(x), and thickness dx. This method is useful when the disk/washer method would be difficult to set up.
A Maclaurin series is a Taylor series centered at a = 0 : ∑(n=0 to ∞) f⁽ⁿ⁾(0)/n! · xⁿ. Common examples include the series for eˣ, sin(x), and cos(x). It is simply the special case of a Taylor series at the origin.
LIATE is a guideline for choosing u in integration by parts. In order of priority: L I A T rigonometric, E