CalcRef

Topics · 19 formulas

Integrals

Fundamental theorem, antiderivatives, and integral properties.

Riemann Sum

∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗)Δx\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x

The definite integral is defined as the limit of Riemann sums. Partition [a,b] into n subintervals of width Δx = (b-a)/n.

Fundamental Theorems

Open formula

FTC Part 1

ddx[∫axf(t) dt]=f(x)\frac{d}{dx}\left[\int_a^x f(t)\, dt\right] = f(x)

The Fundamental Theorem of Calculus Part 1: The derivative of the integral (with variable upper limit) of a continuous function is the original function.

Conditions: f must be continuous on [a, x].

Fundamental Theorems

Open formula

FTC Part 2

∫abf(x) dx=F(b)−F(a) where F′(x)=f(x)\int_a^b f(x)\, dx = F(b) - F(a) \text{ where } F'(x) = f(x)

The Fundamental Theorem of Calculus Part 2: A definite integral can be evaluated using any antiderivative F of f.

Conditions: f must be continuous on [a, b]. F is any antiderivative of f.

Fundamental Theorems

Open formula

Linearity of Integrals

∫ab[αf(x)+βg(x)] dx=α∫abf(x) dx+β∫abg(x) dx\int_a^b [\alpha f(x) + \beta g(x)]\, dx = \alpha \int_a^b f(x)\, dx + \beta \int_a^b g(x)\, dx

Integrals are linear: constants factor out and the integral of a sum is the sum of integrals.

Integral Properties

Open formula

Additivity of Integrals

∫abf(x) dx+∫bcf(x) dx=∫acf(x) dx\int_a^b f(x)\, dx + \int_b^c f(x)\, dx = \int_a^c f(x)\, dx

The integral over [a,c] can be split at any point b into two integrals.

Integral Properties

Open formula

Reversal of Limits

∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\, dx = -\int_b^a f(x)\, dx

Switching the limits of integration negates the integral.

Integral Properties

Open formula

Power Rule for Integration

∫xn dx=xn+1n+1+C(n≠−1)\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

The reverse of the power rule for derivatives. Increase the exponent by 1 and divide by the new exponent.

Conditions: n ≠ -1 (that case gives ln|x|).

Basic Antiderivatives

Open formula

Integral of 1/x

∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\, dx = \ln|x| + C

The antiderivative of 1/x is the natural logarithm of the absolute value of x.

Conditions: x ≠ 0.

Basic Antiderivatives

Open formula

Integral of eˣ

∫ex dx=ex+C\int e^x \, dx = e^x + C

The integral of eˣ is itself, just like its derivative.

Basic Antiderivatives

Open formula

Integral of aˣ

∫ax dx=axln⁡a+C\int a^x \, dx = \frac{a^x}{\ln a} + C

The antiderivative of a general exponential function.

Conditions: a > 0, a ≠ 1.

Basic Antiderivatives

Open formula

Integral of sin(x)

∫sin⁡x dx=−cos⁡x+C\int \sin x \, dx = -\cos x + C

The antiderivative of sine is negative cosine.

Basic Antiderivatives

Open formula

Integral of cos(x)

∫cos⁡x dx=sin⁡x+C\int \cos x \, dx = \sin x + C

The antiderivative of cosine is sine.

Basic Antiderivatives

Open formula

Integral of sec²(x)

∫sec⁡2x dx=tan⁡x+C\int \sec^2 x \, dx = \tan x + C

The antiderivative of secant squared is tangent.

Basic Antiderivatives

Open formula

Integral of csc²(x)

∫csc⁡2x dx=−cot⁡x+C\int \csc^2 x \, dx = -\cot x + C

The antiderivative of cosecant squared is negative cotangent.

Basic Antiderivatives

Open formula

Integral of sec(x)tan(x)

∫sec⁡xtan⁡x dx=sec⁡x+C\int \sec x \tan x \, dx = \sec x + C

The antiderivative of sec(x)tan(x) is sec(x). This is the reverse of the derivative of sec(x).

Basic Antiderivatives

Open formula

Integral of csc(x)cot(x)

∫csc⁡xcot⁡x dx=−csc⁡x+C\int \csc x \cot x \, dx = -\csc x + C

The antiderivative of csc(x)cot(x) is -csc(x).

Basic Antiderivatives

Open formula

Integral yielding arcsin

∫1a2−x2 dx=arcsin⁡(xa)+C\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C

An integral that produces the inverse sine function. Recognizing this pattern is key.

Conditions: |x| < a, a > 0.

Special Integrals

Open formula

Integral yielding arctan

∫1a2+x2 dx=1aarctan⁡(xa)+C\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C

An integral that produces the inverse tangent function.

Conditions: a ≠ 0.

Special Integrals

Open formula

Average Value of a Function

favg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x) \, dx

The average value of f on [a, b] is the integral divided by the interval length.

Special Integrals

Open formula