AP Calculus BC 6.8 Finding Antiderivatives and Indefinite Integrals Study Notes - New Syllabus
AP Calculus BC 6.8 Finding Antiderivatives and Indefinite Integrals Study Notes- New syllabus
AP Calculus BC 6.8 Finding Antiderivatives and Indefinite Integrals Study Notes – AP Calculus BC- per latest AP Calculus BC Syllabus.
LEARNING OBJECTIVE
- Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.
Key Concepts:
- Finding Antiderivatives and Indefinite Integrals
Finding Antiderivatives and Indefinite Integrals
Finding Antiderivatives and Indefinite Integrals
In this topic, we Study about the reverse process of differentiation, known as antidifferentiation. This process helps us find a function whose derivative is a given function. The result is called an indefinite integral.
If \( F'(x) = f(x) \), then \( F(x) \) is called an antiderivative of \( f(x) \), and we write:
\( \displaystyle \int f(x)\,dx = F(x) + C \)
where \( C \) is the constant of integration, accounting for the family of all antiderivatives.
Basic Rules of Indefinite Integration:
| Rule | Description |
|---|---|
| \( \displaystyle \int k\,dx = kx + C \) | The integral of a constant \( k \) |
| \( \displaystyle \int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C \), \( n \ne -1 \) | Power rule for integration |
| \( \displaystyle \int \dfrac{1}{x}\,dx = \ln|x| + C \) | Special case when \( n = -1 \) |
| \( \displaystyle \int e^x\,dx = e^x + C \) | Exponential function base \( e \) |
| \( \displaystyle \int a^x\,dx = \dfrac{a^x}{\ln a} + C \) | Exponential function with base \( a > 0 \), \( a \ne 1 \) |
| \( \displaystyle \int \sin x\,dx = -\cos x + C \) | Integral of sine |
| \( \displaystyle \int \cos x\,dx = \sin x + C \) | Integral of cosine |
Example:
Find the indefinite integral: \( \displaystyle \int (3x^2 – 4x + 5)\,dx \)
▶️Answer/Explanation
Integrate term by term using the power rule:
\( \displaystyle \int 3x^2\,dx = x^3 \)
\( \displaystyle \int -4x\,dx = -2x^2 \)
\( \displaystyle \int 5\,dx = 5x \)
Final answer: \( x^3 – 2x^2 + 5x + C \)
Example:
Evaluate \( \displaystyle \int (e^x + \cos x – \dfrac{1}{x})\,dx \)
▶️Answer/Explanation
Use known integrals:
\( \displaystyle \int e^x\,dx = e^x \)
\( \displaystyle \int \cos x\,dx = \sin x \)
\( \displaystyle \int \dfrac{1}{x}\,dx = \ln|x| \)
So the result is: \( e^x + \sin x – \ln|x| + C \)
Example:
Evaluate: \( \displaystyle \int (4x^3 + 7x – 9)\,dx \)
▶️Answer/Explanation
Use the linearity property: integrate term-by-term.
\( \displaystyle \int 4x^3\,dx = x^4 \)
\( \displaystyle \int 7x\,dx = \dfrac{7}{2}x^2 \)
\( \displaystyle \int -9\,dx = -9x \)
Final answer: \( x^4 + \dfrac{7}{2}x^2 – 9x + C \)
Example:
If \( \displaystyle \int f(x)\,dx = F(x) + C \), and \( \displaystyle \int g(x)\,dx = G(x) + C \), then what is \( \displaystyle \int (2f(x) – 3g(x))\,dx \)?
▶️Answer/Explanation
Use linearity of integrals: constants factor out and integrals distribute.
\( \displaystyle \int (2f(x) – 3g(x))\,dx = 2\displaystyle \int f(x)\,dx – 3\displaystyle \int g(x)\,dx \)
= \( 2F(x) – 3G(x) + C \)
(Remember: all antiderivatives are up to a constant, so we combine constants into a single \( C \)).
Example:
Find \( \displaystyle \int (5\cos x – 2\sin x + 6)\,dx \)
▶️Answer/Explanation
Use basic integration rules:
\( \displaystyle \int 5\cos x\,dx = 5\sin x \)
\( \displaystyle \int -2\sin x\,dx = 2\cos x \)
\( \displaystyle \int 6\,dx = 6x \)
Final answer: \( 5\sin x + 2\cos x + 6x + C \)
Example:
Evaluate \( \displaystyle \int \left( \dfrac{1}{x} + 7x^{-3} \right) dx \)
▶️Answer/Explanation
Recall:
- \( \displaystyle \int \dfrac{1}{x} dx = \ln|x| \)
- \( \displaystyle \int x^n dx = \dfrac{x^{n+1}}{n+1} \) for \( n \ne -1 \)
So:
\( \displaystyle \int \left( \dfrac{1}{x} + 7x^{-3} \right) dx = \ln|x| + 7 \cdot \dfrac{x^{-2}}{-2} + C \)
= \( \ln|x| – \dfrac{7}{2x^2} + C \)