Indefinite Integral - Basic Integration Rules, Problems, Formulas, Trig Functions, Calculus
The Organic Chemistry Tutor・2 minutes read
The antiderivatives of various functions, such as the integral of x squared or cube root of x to the fourth, follow specific patterns and rules that yield simplified results by integrating variable powers or trigonometric functions using techniques like u-substitution or integration by parts. Understanding these integral rules and methods allows for the efficient calculation of antiderivatives for various types of functions, simplifying complex expressions to concise forms.
Insights
- The process of finding antiderivatives involves adding a variable to a constant, as seen in examples like the integral of pi dy, e dz, and x squared dx, where the result includes the variable being integrated and a constant term.
- Various integration techniques such as u-substitution, integration by parts, and trigonometric substitution are essential for solving complex integrals, as demonstrated in cases like tan(x), x * cos(x) dx, 4 / (1 + x^2) dx, and √(1 - x^2) dx, showcasing the versatility and power of these methods in calculus.
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Recent questions
How do you find the antiderivative of a constant?
By integrating a constant, the result is the constant times x plus c, where c is the constant of integration. This process involves adding x to the constant value to obtain the antiderivative.
What is the antiderivative of e dz?
The antiderivative of e dz simplifies to e times z plus c. This showcases that when integrating a constant like e, the result involves multiplying the constant by the variable and adding a constant of integration.
How do you integrate a variable raised to a constant?
Integrating a variable raised to a constant, such as x raised to the n, results in x raised to the n plus one divided by n plus one plus c. This pattern demonstrates how to integrate variable powers by increasing the exponent and dividing by the new exponent.
What is the antiderivative of x squared dx?
The antiderivative of x squared dx simplifies to x to the third divided by three plus c. This follows the pattern of integrating variable powers, where the exponent increases by one and is divided by the new exponent.
How do you find the integral of √(1 - x^2) dx?
To solve the integral of √(1 - x^2) dx, use trigonometric substitution by setting x = sin(θ) to simplify the expression to 3θ + c, then replace θ with inverse sin(x) to get 3 * arcsin(x) + c. This method allows for the integration of square root functions involving trigonometric identities.
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Summary
00:00
Integrating Functions: Rules and Examples
- The integral of 4 dx is 4x plus c, as the antiderivative of a constant involves adding x and a constant value.
- The antiderivative of pi dy is pi times y plus c, showcasing how adding a variable to a constant yields the antiderivative.
- For the antiderivative of e dz, the result is e times z plus c, emphasizing that e as a constant simplifies the integration.
- Integrating a variable raised to a constant, like x raised to the n, results in x raised to the n plus one divided by n plus one plus c.
- The antiderivative of x squared dx is x to the third divided by three plus c, following the pattern of integrating variable powers.
- When dealing with expressions like 8x cubed, the antiderivative involves adding one to the exponent and dividing by the result, simplifying to 2x to the fourth.
- Integrating a polynomial function like x squared minus five x plus six requires integrating each term separately, yielding x to the third divided by three, 5x squared divided by 2, and adding a variable for the constant.
- The antiderivative of the square root of x is two-thirds square root x cubed plus c, showcasing the process of integrating square root functions.
- For the cube root of x to the fourth, the antiderivative simplifies to three over seven cube root x to the seventh power plus c, demonstrating the integration of cube root functions.
- The antiderivative of 3x minus 1 squared dx involves foiling the expression, simplifying to 3x cubed divided by three minus 3x squared divided by two minus 2x plus c.
18:27
Integrals: Techniques and Solutions
- To find the integral of 5x^u * u^4 * du / 2x, cancel out x and rearrange to get 5/2 ∫ u^4 du, then apply the power rule to get 1/2 * u^5 + c, which simplifies to 1/2 * (x^2 + 3)^5 + c.
- For the integral of tan(x), use u-substitution by setting u = cos(x) to simplify the expression to -ln(cos(x)), then rearrange to ln(sec(x)) + c.
- When dealing with x * cos(x) dx, apply integration by parts with u = x and dv = cos(x) dx, leading to the final result of x * sin(x) - cos(x) + c.
- To solve the integral of 4 / (1 + x^2) dx, use trigonometric substitution by setting x = tan(θ) to simplify the expression to 4θ + c, then replace θ with inverse tan(x) to get 4 * arctan(x) + c.
- For the integral of √(1 - x^2) dx, use trigonometric substitution by setting x = sin(θ) to simplify the expression to 3θ + c, then replace θ with inverse sin(x) to get 3 * arcsin(x) + c.
