Calculus formulas

$x$ in radians, $a>0$. Each follows from the Maclaurin series of the function.

The last holds for any real $n$ when $a>0$.

For $p>0$ and $a>0$: a logarithm grows more slowly than any positive power, and a power more slowly than any growing exponential. Each follows from L'Hospital's rule. Consequence: ${\lim }_{x\to 0^{+}}{x}^x={\lim }_{x\to 0^{+}}{e}^{x\ln x}=e^0=1$.

Valid whenever the limit in the exponent exists; if that limit is $+\infty$ the answer is $\infty$, and if it is $-\infty$ the answer is $0$. It comes from $\ln f\approx f-1$ when $f\to 1$.

Only for the forms $\frac{0}{0}$ or $\frac{\infty }{\infty }$, with $f,g$ differentiable and $g^{\prime }(x)\ne 0$ near $a$ (except possibly at $a$), and only if $\lim f^{\prime }(x)/g^{\prime }(x)$ exists (finite or infinite). Differentiate numerator and denominator separately, not by the quotient rule. Also valid for $a=\pm \infty$ and for one-sided limits.

Continuity needs $f(a)$ defined, the limit to exist, and the two to be equal. Differentiability needs the left derivative ($h\to 0^{-}$) and the right derivative ($h\to 0^{+}$) to be finite and equal. Differentiable $\Rightarrow$ continuous; the converse is false.

Geometrically, a horizontal tangent between two equal heights. $f(x)=|x|$ on $[-1,1]$ has $f(-1)=f(1)$ but no such $c$, because it is not differentiable at 0.

Needs $f$ continuous on $[a,b]$ and differentiable on $(a,b)$. The tangent at $c$ is parallel to the chord. Rolle's theorem is the case $f(a)=f(b)$. Equivalent form: $f(b)=f(a)+(b-a)f^{\prime }(c)$. Consequences on an interval: $f^{\prime }=0$ throughout $\Rightarrow$ $f$ is constant; $f^{\prime }>0$ $\Rightarrow$ $f$ is strictly increasing; $f^{\prime }<0$ $\Rightarrow$ strictly decreasing; $|f^{\prime }|\le M$ $\Rightarrow$ $|f(b)-f(a)|\le M(b-a)$.

Needs $f,g$ continuous on $[a,b]$, differentiable on $(a,b)$, and $g^{\prime }(x)\ne 0$ on $(a,b)$ (which also forces $g(b)\ne g(a)$). Lagrange's theorem is the case $g(x)=x$.

Maclaurin series when $a=0$. Needs derivatives of every order at $a$, and the series equals $f(x)$ exactly when the remainder $R_n(x)\to 0$; for the standard functions below this holds throughout the stated range. A convergent Taylor series can still differ from $f$: $f(x)=e^{-1/x^2}$ with $f(0)=0$ has every derivative zero at 0, so its Maclaurin series is 0. A power series that equals $f$ near $a$ is its Taylor series, so series may be built by substitution, multiplication, and term-by-term differentiation or integration inside the interval of convergence.

Odd functions have only odd powers, even functions only even powers.

For a non-negative integer $k$ the binomial series stops after $k+1$ terms and holds for all $x$. Replacing $x$ by $-x$ gives $\ln (1-x)=-x-\frac{x^2}{2}-\frac{x^3}{3}-\cdots$ for $-1\le x<1$, and $\frac{1}{1+x}=1-x+x^2-\cdots$ for $|x|<1$.

Needs $f$ continuous on $[a,b]$. Evaluating an antiderivative across a point where $f$ is unbounded is not justified and can give a wrong answer (see the tip at the end of this section): split the integral there and treat each piece as improper.

$u$ and $v$ with continuous derivatives on $[a,b]$. Choose as $u$ the factor that simplifies when differentiated (a power of $x$, a logarithm, an inverse trigonometric function). The last two lines come from integrating by parts twice; $a^2+b^2\ne 0$ and $C$ is an arbitrary constant.

Substitute $x\to a+b-x$. Adding the two forms often cancels the awkward part: $2I={\int }_a^b[f(x)+f(a+b-x)]dx$.

Even: $f(-x)=f(x)$; odd: $f(-x)=-f(x)$. Any $f$ is its even part $\frac{f(x)+f(-x)}{2}$ plus its odd part $\frac{f(x)-f(-x)}{2}$, and only the even part survives a symmetric integral.

Example: ${\int }_0^{\pi }{\sin }^{n}xdx=2{\int }_0^{\pi /2}{\sin }^{n}xdx$, and ${\int }_0^{\pi }{\cos }^{n}xdx=0$ for odd $n$.

For $f(x+T)=f(x)$ and a positive integer $n$: every full period gives the same integral. Example: $|\sin x|$ has period $\pi$, so ${\int }_0^{10\pi }|\sin x|dx=10{\int }_0^{\pi }\sin xdx=20$.

$n$ a positive integer. Each product keeps only its positive factors, so it runs down to 1 or 2; a product with no factors is 1, so $n=1$ gives 1. Examples: $n=4$ gives $\frac{3\cdot 1}{4\cdot 2}\cdot \frac{\pi }{2}=\frac{3\pi }{16}$; $n=5$ gives $\frac{4\cdot 2}{5\cdot 3\cdot 1}=\frac{8}{15}$.

$m,n$ non-negative integers. Each product keeps only its positive factors, running down to 1 or 2, and a product with no factors (for $m$ or $n$ equal to 0 or 1) is 1. With $m=0$ it is the formula above. Examples: $m=n=2$ gives $\frac{1\cdot 1}{4\cdot 2}\cdot \frac{\pi }{2}=\frac{\pi }{16}$; $m=3,n=2$ gives $\frac{2\cdot 1}{5\cdot 3\cdot 1}=\frac{2}{15}$. The Beta function below gives the same values.

Needs $f$ and $\partial f/\partial x$ continuous and $a(x),b(x)$ differentiable. If the integrand does not contain $x$ the last term is zero; if the limits are constants only the last term remains (differentiation under the integral sign).

Both diverge at $p=1$ (logarithm). For non-negative $f$ and $g$: comparison test, if $0\le f\le g$ then $\int g$ convergent $\Rightarrow$ $\int f$ convergent, and $\int f$ divergent $\Rightarrow$ $\int g$ divergent; limit comparison, if $f/g$ tends to a finite non-zero limit, both converge or both diverge. For any $f$: if $\int |f|$ converges, so does $\int f$ (absolute convergence).

$a>0$, $b$ real and $n$ a non-negative integer; for real $n>-1$ the second result is $\frac{\Gamma (n+1)}{a^{n+1}}$. The last integral converges, but not absolutely.

$\Gamma (n+1)=n!$ for non-negative integers $n$, so $\Gamma (1)=1$. Example: $\Gamma \left(\frac{5}{2}\right)=\frac{3}{2}\cdot \frac{1}{2}\sqrt{\pi }=\frac{3\sqrt{\pi }}{4}$.

$m,n>0$ and $B(m,n)=B(n,m)$. The trigonometric form gives ${\int }_0^{\pi /2}{\sin }^p\theta {\cos }^q\theta d\theta =\frac{1}{2}B\left(\frac{p+1}{2},\frac{q+1}{2}\right)$ for $p,q>-1$.

Area between two curves and area swept by a polar curve $r(\theta )$; arc length for $y=f(x)$, for a parametric curve, and for a polar curve. $V$ is the volume of the solid formed by revolving the region under $y=f(x)$, $a\le x\le b$, about the $x$-axis; $V_y$ (shell method, for $f\ge 0$ and $0\le a<b$) is the volume when the same region is revolved about the $y$-axis.

For $z=f(x,y)$ with $x=x(t)$, $y=y(t)$. For $x=x(u,v)$, $y=y(u,v)$: $z_u=z_x{x}_u+z_y{y}_u$ and $z_v=z_x{x}_v+z_y{y}_v$. The total differential also estimates small changes: $\Delta z\approx z_x\Delta x+z_y\Delta y$.

Valid where $f_y\ne 0$. Note the minus sign.

All derivatives are evaluated at $(a,b)$, and $f$ needs continuous partial derivatives of every order used. Stopping after the first bracket gives the linear approximation $f(a+h,b+k)\approx f(a,b)+h{f}_x+k{f}_y$, the total differential again. At a stationary point the first bracket is zero and the second decides the nature of the point, which is where the $rt-s^2$ test comes from.

For $f$ differentiable and homogeneous of degree $n$, meaning $f(tx,ty)=t^{n}f(x,y)$ for all $t>0$ (continuous second partial derivatives are needed for the second identity). $n$ may be negative or fractional. With more variables: ${\sum }_i{x}_i\partial f/\partial x_i=nf$.

Classify with the table below; $f$ needs continuous second partial derivatives near the point. When $D>0$, $r$ and $t$ have the same sign, so either may be used.

Extremise $f(x,y,z)$ subject to the constraint $g(x,y,z)=0$, at points where $\nabla g\ne 0$. Solve the four equations for $x,y,z,\lambda$, evaluate $f$ at each candidate and compare. The method finds candidates; it does not by itself say which is a maximum.

Area of a plane region $R$, volume under the surface $z=f(x,y)$ over $R$, and volume of a solid $E$. Replacing 1 by a density gives mass.

Valid when the map $(u,v)\mapsto (x,y)$ takes $R^{\prime }$ one-to-one onto $R$ (except possibly on the boundary), with continuous partial derivatives and a non-zero Jacobian inside. The absolute value of the Jacobian is the area scale factor. The Jacobians of a transformation and its inverse multiply to 1: $\frac{\partial (x,y)}{\partial (u,v)}\cdot \frac{\partial (u,v)}{\partial (x,y)}=1$.

The first follows by squaring the integral and switching to polar coordinates: ${\left({\int }_{-\infty }^{\infty }{e}^{-x^2}dx\right)}^2={\int }_0^{2\pi }{\int }_0^{\infty }{e}^{-r^2}rdrd\theta =\pi$ (with $a,b,c>0$ in the others).

Period $2L$. For period $2\pi$ put $L=\pi$, giving $a_n\cos nx+b_n\sin nx$.

Any interval of length $2L$, such as $(0,2L)$, may be used. They follow from orthogonality: over one period, $\cos \frac{m\pi x}{L}\cos \frac{n\pi x}{L}$ and $\sin \frac{m\pi x}{L}\sin \frac{n\pi x}{L}$ integrate to $L$ if $m=n\ge 1$ and to $0$ if $m\ne n$, and every sine-cosine product integrates to $0$.

A function given only on $(0,L)$ is extended evenly (cosine series) or oddly (sine series), each with period $2L$. Both converge to $f$ at every point of $(0,L)$ where $f$ is continuous. They differ outside $(0,L)$ and can differ at its end points: the sine series is $0$ at $x=0$ and $x=L$, while the cosine series converges to $f(0^{+})$ at $x=0$ and to $f(L^{-})$ at $x=L$.

Written for the $\frac{a_0}{2}$ convention; holds whenever $\int f^2$ over a period is finite, in particular under the Dirichlet conditions above. Halving both sides: mean square value $={\left(\frac{a_0}{2}\right)}^2+\frac{1}{2}{\sum }_{n\ge 1}(a_n^2+b_n^2)$.