Power Rule for Integrals
TLDRThe video script offers a step-by-step guide to evaluating five different integrals using fundamental calculus formulas. It covers the integral of a constant multiplied by dx, the power rule for integration, and handling roots and negative exponents. Each integral is meticulously broken down, applying the appropriate rules to arrive at the final answer, which includes the constant of integration. The explanations are clear, making complex integration problems accessible and understandable.
Takeaways
- 📚 The integral of 3dx is 3x + C, using the basic property of constants multiplying the integral function.
- 🔢 For the integral of x^3 dx, apply the power rule, resulting in (x^(3+1))/(3+1) + C, which simplifies to (x^4)/4 + C.
- 🌟 When integrating 2√x dx, first recognize √x as x^(1/2) and apply the power rule, yielding 2x^(3/2)/3 + C.
- 🔄 For the integral of dx/x^4, transform the expression into negative exponent form, 1/x^(-4), and apply the power rule to get -1/3x^(-3) + C, or -1/(3x^3) + C.
- 🧮 The integral of 2dx/(∛x), after factoring out the constant and expressing ∛x as x^(1/3), becomes 2∫x^(-1/3)dx, which simplifies to (3/2)x^(2/3) + C.
- 📈 The power rule is a fundamental tool for evaluating integrals, especially when dealing with x raised to a power.
- 🔧 Constants can be factored out of integrals, which simplifies the process and the final result.
- 🌐 Understanding fractional and negative exponents is crucial for transforming and solving complex integrals.
- 🧠 The process of integrating involves recognizing patterns, applying rules, and simplifying the expressions to reach the final answer.
- 📊 Each integral has a unique approach based on the function's form, emphasizing the importance of adaptability in problem-solving.
Q & A
What is the basic formula for integrating dx?
-The basic formula for integrating dx is ∫dx = x + C, where C is the constant of integration.
How do you evaluate the integral of a constant multiplied by dx?
-To evaluate the integral of a constant multiplied by dx, such as ∫3dx, you simply multiply the constant by the integral of dx. So, ∫3dx = 3 * (x + C) = 3x + 3C.
What is the power rule for integration and how is it applied?
-The power rule for integration states that ∫x^n dx = x^(n+1)/(n+1) + C, provided that n is not equal to -1. It is applied by replacing n with the exponent of x in the integrand.
How do you evaluate the integral of x^3 dx?
-To evaluate the integral of x^3 dx, apply the power rule with n=3. So, ∫x^3 dx = x^(3+1)/ (3+1) + C = x^4/4 + C.
What is the method to integrate a function with a radical like the square root of x?
-To integrate a function with a radical, like √x, first rewrite the radical as a fractional exponent. So, √x is the same as x^(1/2). Then, apply the power rule with the new exponent to find the integral.
How do you simplify the expression 2 integrals of x^(1/2) dx?
-The expression 2 integrals of x^(1/2) dx simplifies to 2 * (x^(1/2 + 1) / (1/2 + 1) + C). After simplifying, this becomes 2 * x^(3/2) / (3/2) + C, which further simplifies to (4/3)x^(3/2) + C.
What happens when you encounter a negative exponent in integration?
-When you encounter a negative exponent in integration, you rewrite it as the reciprocal of the positive exponent. For example, x^(-4) becomes 1/x^4. Then, you apply the power rule with the new exponent.
How do you evaluate the integral of dx/x^4?
-To evaluate the integral of dx/x^4, first rewrite x^(-4) as 1/x^4. Then, apply the power rule with n=-4. So, ∫dx/x^4 = -1/x^3 + C, or equivalently, -x^(-3) + C.
What is the process for integrating a function with a cube root in the denominator?
-For integrating a function with a cube root in the denominator, like ∫2dx/(3√x), first rewrite the cube root as a fractional exponent, x^(1/3). Then, apply the power rule with the new exponent, and simplify the resulting expression.
How do you simplify the expression 2 integrals of dx/(3√x)?
-The expression 2 integrals of dx/(3√x) simplifies to 2 * (x^(1/3 + 1) / (1/3 + 1) + C). After simplifying, this becomes (2/3) * x^(2/3) + C, which can be further simplified to (2/3)x^(2/3) + C.
What is the final result of evaluating the integral ∫2dx/(3√x)?
-The final result of evaluating the integral ∫2dx/(3√x) is (3/2)x^(2/3) + C.
Outlines
📚 Integral Evaluation Techniques
This paragraph introduces the process of evaluating integrals using fundamental formulas. It explains how to evaluate the integral of 3dx using the basic integral formula for dx, resulting in 3x + C. The paragraph then proceeds to evaluate the integral of x^3 dx using the power rule, yielding x^(3+1)/(3+1) + C. It also covers the integral of 2√x dx by first rewriting the square root as a fractional exponent and then applying the power rule, leading to 2x^(1/2+1)/(1/2+1) + C. Lastly, it discusses the integral of dx/x^4 by transforming the negative exponent into a positive one and applying the power rule, resulting in -1/(3x^3) + C.
🔢 Solving Integrals with Fractional Exponents
This paragraph focuses on evaluating integrals involving fractional exponents, specifically dealing with the integral of 2dx over the cube root of x. The constant 2 is factored out, and the cube root is transformed into a fractional exponent (x^(1/3)). The integral is then evaluated by applying the power rule to x^(-1/3), resulting in 2 * (x^(-1/3 + 1))/(-1/3 + 1) + C. The final step involves simplifying the complex fraction to obtain the result of 3/2 * x^(2/3) + C.
Mindmap
Keywords
💡Integral
💡Power Rule
💡Constant of Integration
💡Fractional Exponents
Highlights
The integral of 3dx is evaluated using the basic integration formula.
The integral of dx is equal to x plus the constant of integration, c.
Constants can be factored out of the integral, resulting in 3 times the integral of dx.
The integral of x cubed dx is evaluated using the power rule.
For x raised to the power of n, the integral is x raised to n+1 over n+1 plus c, with n not equal to -1.
The integral of 2 square root of x dx is evaluated by applying the power rule to the fractional exponent.
Square root of x is equivalent to x raised to the power of 1/2.
The integral of dx over x to the fourth is transformed into a negative exponent before applying the power rule.
Negative exponents are handled by taking the reciprocal of the base raised to the positive exponent.
The integral of 2 dx over the cube root of x is evaluated by converting the cube root to a fractional exponent.
Cube root of x is the same as x raised to the power of 1/3.
The power rule is applied to x raised to the power of -1/3 to find the integral.
The final answer for the integral of 3dx is 3x plus c.
The final answer for the integral of x cubed dx is x to the power of 4/3 plus c.
The final answer for the integral of 2 square root of x dx is 4/3 * x to the power of 3/2 plus c.
The final answer for the integral of dx over x to the fourth is -1/3 * x cubed plus c.
The final answer for the integral of 2 dx over the cube root of x is 3 * x to the power of 2/3 plus c.
Integration formulas and power rules are essential tools for solving integrals.
Understanding the properties of exponents, including negative and fractional exponents, is crucial for evaluating integrals.
Transcripts
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