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bookIntroduction to Integrals

Integration is a fundamental concept in calculus that helps us compute the total accumulation of a quantity, such as areas under curves. It plays a key role in data science, from probability distributions to optimization.

What is an Integral?

An integral is the mathematical process of finding the total accumulation of a quantity. You can think of it as finding the area under a curve. If differentiation gives the rate of change, integration reconstructs the original function.

The basic integral of a power function follows this rule:

∫Cxndx=C(xn+1n+1)+C\int Cx^ndx = C\left( \frac{x^{n+1}}{n+1} \right) + C

Where:

  • CC is a constant;
  • n≠−1n \neq -1;
  • ...+C...+C represents an arbitrary constant of integration.

Key idea: if differentiation reduces the power of xx, integration increases it.

Common Integral Rules

Power Rule for Integration

This rule helps integrate any polynomial expression:

∫xndx=xn+1n+1+C, n≠−1\int x^ndx = \frac{x^{n+1}}{n+1}+ C,\ n \neq -1

Example:

∫x2dx=x33+C\int x^2dx = \frac{x^3}{3}+C

Exponential Rule

The integral of the exponential function exe^x is unique because it remains the same after integration:

∫exdx=ex+C\int e^xdx = e^x + C

But if we have an exponent with a coefficient, we use another rule:

∫eaxdx=1aeax+C, a≠0\int e^{ax}dx = \frac{1}{a}e^{ax}+C,\ a \neq 0

Example:

∫e2xdx=e2x2+C\int e^{2x}dx = \frac{e^{2x}}{2} + C

Trigonometric Integrals

Sine and cosine functions also follow straightforward integration rules:

∫sin(x)dx=−cos(x)+C∫cos(x)dx=sin(x)+C\int sin(x)dx = -cos(x) + C \\ \int cos(x)dx = sin(x) + C

Definite Integrals

Unlike indefinite integrals, which include an arbitrary constant CC, definite integrals evaluate a function between two limits aa and bb:

∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx = F(b) - F(a)

Where F(x)F(x) is the antiderivative of f(x)f(x).

Example:

∫022x dx=[x2]=4−0=4\int^2_0 2x\ dx = \left[ x^2 \right] = 4 - 0 = 4

This means the area under the curve y=2xy = 2x from x=0x=0 to x=2x=2 is 44.

1. Calculate the integral:

∫3x2dx\int 3x^2 dx

2. Calculate the integral:

∫e3xdx\int e^{3x} dx
question mark

Calculate the integral:

∫3x2dx\int 3x^2 dx

Select the correct answer

question mark

Calculate the integral:

∫e3xdx\int e^{3x} dx

Select the correct answer

War alles klar?

Wie können wir es verbessern?

Danke für Ihr Feedback!

Abschnitt 3. Kapitel 5

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bookIntroduction to Integrals

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Integration is a fundamental concept in calculus that helps us compute the total accumulation of a quantity, such as areas under curves. It plays a key role in data science, from probability distributions to optimization.

What is an Integral?

An integral is the mathematical process of finding the total accumulation of a quantity. You can think of it as finding the area under a curve. If differentiation gives the rate of change, integration reconstructs the original function.

The basic integral of a power function follows this rule:

∫Cxndx=C(xn+1n+1)+C\int Cx^ndx = C\left( \frac{x^{n+1}}{n+1} \right) + C

Where:

  • CC is a constant;
  • n≠−1n \neq -1;
  • ...+C...+C represents an arbitrary constant of integration.

Key idea: if differentiation reduces the power of xx, integration increases it.

Common Integral Rules

Power Rule for Integration

This rule helps integrate any polynomial expression:

∫xndx=xn+1n+1+C, n≠−1\int x^ndx = \frac{x^{n+1}}{n+1}+ C,\ n \neq -1

Example:

∫x2dx=x33+C\int x^2dx = \frac{x^3}{3}+C

Exponential Rule

The integral of the exponential function exe^x is unique because it remains the same after integration:

∫exdx=ex+C\int e^xdx = e^x + C

But if we have an exponent with a coefficient, we use another rule:

∫eaxdx=1aeax+C, a≠0\int e^{ax}dx = \frac{1}{a}e^{ax}+C,\ a \neq 0

Example:

∫e2xdx=e2x2+C\int e^{2x}dx = \frac{e^{2x}}{2} + C

Trigonometric Integrals

Sine and cosine functions also follow straightforward integration rules:

∫sin(x)dx=−cos(x)+C∫cos(x)dx=sin(x)+C\int sin(x)dx = -cos(x) + C \\ \int cos(x)dx = sin(x) + C

Definite Integrals

Unlike indefinite integrals, which include an arbitrary constant CC, definite integrals evaluate a function between two limits aa and bb:

∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx = F(b) - F(a)

Where F(x)F(x) is the antiderivative of f(x)f(x).

Example:

∫022x dx=[x2]=4−0=4\int^2_0 2x\ dx = \left[ x^2 \right] = 4 - 0 = 4

This means the area under the curve y=2xy = 2x from x=0x=0 to x=2x=2 is 44.

1. Calculate the integral:

∫3x2dx\int 3x^2 dx

2. Calculate the integral:

∫e3xdx\int e^{3x} dx
question mark

Calculate the integral:

∫3x2dx\int 3x^2 dx

Select the correct answer

question mark

Calculate the integral:

∫e3xdx\int e^{3x} dx

Select the correct answer

War alles klar?

Wie können wir es verbessern?

Danke für Ihr Feedback!

Abschnitt 3. Kapitel 5
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