Elementary antiderivatives, coefficients and domains
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| antiderivative/ˌæntɪdɪˈrɪvətɪv/ | 原函数 | yuán hán shù |
Differentiation multiplies an exponential or trigonometric argument by its coefficient. How must integration undo that factor?
- Differentiation multiplies an exponential or trigonometric argument by its coefficient. How must integration undo that factor?
- This lesson studies antiderivative 原函数: A function whose derivative is the specified integrand on the stated interval.
Choose the mathematical structure
- For real powers on an allowed differentiable domain, integrate x^n as x^(n+1)/(n+1)+C when n≠−1. Integrate 1/x as ln|x|+C on an interval excluding zero. For k≠0, integrals of e^(kx), sin(kx), cos(kx) are e^(kx)/k, −cos(kx)/k, sin(kx)/k respectively, plus constants. Apply linearity to sums and constant multiples; verify by differentiation.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines antiderivative?
A function whose derivative is the specified integrand on the stated interval.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For f(x)=6x²−4e^(2x)+3cos(3x)−2sin(2x)+5/x with x>0, F=2x³−2e^(2x)+sin(3x)+cos(2x)+5ln x+C. Each derivative recovers its matching term. On x<0, integrate 5/x as 5ln(−x), or 5ln|x|; constants can differ on disconnected intervals. The integral of 3cos(3x) from 0 to π/6 is sin(π/2)−sin 0=1. For F′=6x² and F(1)=7, F=2x³+5. If k=0, e^(kx)=1 and cos(kx)=1 integrate to x+C, while sin(kx)=0 integrates to C; never divide by k=0.
Elementary antiderivatives, coefficients and domains
For real powers on an allowed differentiable domain, integrate x^n as x^(n+1)/(n+1)+C when n≠−1
Check the hypothesis, endpoints and coefficients behind each integral calculation.
For F′=6x² and F(1)=7, find the integration constant.
7=2×1³+C gives C=5.
Test a tempting shortcut
- The n=−1 power is a separate logarithmic case, not division by zero in the power formula. Keep the sign in the sine antiderivative and divide by the argument coefficient. ln x alone does not cover negative x. An indefinite integral needs a constant; a definite integral has a numerical value when valid limits are supplied. A real fractional power may restrict the domain.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Integrating cos(kx) always gives k sin(kx), including k=0. This claim is false. Explain which definition or assumption it violates.
Find the worked definite integral of 3cos(3x).
The antiderivative sin(3x) gives sin(π/2)−sin 0=1.
Integrating cos(kx) always gives k sin(kx), including k=0.
The n=−1 power is a separate logarithmic case, not division by zero in the power formula. Keep the sign in the sine antiderivative and divide by the argument coefficient. ln x alone does not cover negative x. An indefinite integral needs a constant; a definite integral has a numerical value when valid limits are supplied. A real fractional power may restrict the domain.
Interpret a new situation
- Differentiate term by term to check coefficients and signs, and state the interval before using logarithms or fractional powers. The H2 named elementary forms are powers, e^(kx), 1/x, sin(kx) and cos(kx). Other forms require separate justification and are not asserted as additional compulsory H2 forms.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Evaluate the derivative of 5ln|x| at x=−2.
The derivative is 5/x on either allowed interval, giving −5/2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · H. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A function whose derivative is the specified integrand on the stated interval. Choose the relationship, show the method, check its assumptions and interpret the result.