Edexcel IAL - Pure Maths 1- 5.2 Integration of xn and Related Expressions- Study notes - New syllabus
Edexcel IAL – Pure Maths 1- 5.2 Integration of xn and Related Expressions -Study notes- New syllabus
Edexcel IAL – Pure Maths 1- 5.2 Integration of xn and Related Expressions -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- Integration of xn and Related Expressions
Integration of $\rm {x^n}$ and Related Forms
Indefinite integration allows us to reverse differentiation. When integrating powers of \( x \), we apply the power rule, except when \( n = -1 \).
General Rule (for \( n \ne -1 \))
\( \displaystyle \int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C \)
Integration works term-by-term and constant multiples are carried outside the integral.
Integration Rules (Sums, Differences, Multiples)
| Rule | Meaning / Formula |
| Constant Multiple Rule | \( \displaystyle \int kf(x)\,dx = k\int f(x)\,dx \) |
| Sum Rule | \( \displaystyle \int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx \) |
| Difference Rule | \( \displaystyle \int [f(x) – g(x)]\,dx = \int f(x)\,dx – \int g(x)\,dx \) |
| Power Rule | \( \displaystyle \int x^n dx = \dfrac{x^{n+1}}{n+1} + C \quad (n\ne -1) \) |
1. Constant Multiple Rule
\( \displaystyle \int kf(x)\,dx = k\int f(x)\,dx \)
This rule states that a constant multiplying a function may be taken outside the integral. It means the integral scales in the same way as the function.
2. Sum Rule
\( \displaystyle \int \left[f(x) + g(x)\right]dx = \int f(x)\,dx + \int g(x)\,dx \)
This rule allows us to integrate each term separately when several terms are added together. Integration distributes over addition.
3. Difference Rule
\( \displaystyle \int \left[f(x) – g(x)\right]dx = \int f(x)\,dx – \int g(x)\,dx \)
This means integration also distributes over subtraction. We can integrate each part individually and subtract afterwards.
4. Power Rule (for \( n \ne -1 \))
\( \displaystyle \int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C \)
This rule reverses the differentiation of \( x^n \). It applies to all powers except \( x^{-1} \), which leads to a logarithmic result.
Example
Integrate:
\( \displaystyle \int \left( \dfrac12 x^2 – 3x^{-1} \right)\,dx \)
▶️ Answer / Explanation
\( \displaystyle \int \dfrac12 x^2\,dx = \dfrac12 \cdot \dfrac{x^3}{3} = \dfrac{x^3}{6} \)
\( \displaystyle \int -3x^{-1}\,dx = -3\ln|x| \)
Final answer:
\( \displaystyle \dfrac{x^3}{6} – 3\ln|x| + C \)
Example
Integrate:
\( \displaystyle \int \dfrac{(x+2)^2}{\sqrt{x}}\,dx \)
▶️ Answer / Explanation
Expand numerator:
\( (x+2)^2 = x^2 + 4x + 4 \)
Divide each term by \( x^{1/2} \):
- \( x^2 \div x^{1/2} = x^{3/2} \)
- \( 4x \div x^{1/2} = 4x^{1/2} \)
- \( 4 \div x^{1/2} = 4x^{-1/2} \)
Integrate term-by-term:
\( \displaystyle \int x^{3/2}\,dx = \dfrac{2}{5}x^{5/2} \)
\( \displaystyle \int 4x^{1/2}\,dx = \dfrac{8}{3}x^{3/2} \)
\( \displaystyle \int 4x^{-1/2}\,dx = 8x^{1/2} \)
Final answer:
\( \displaystyle \dfrac{2}{5}x^{5/2} + \dfrac{8}{3}x^{3/2} + 8x^{1/2} + C \)
Example
Find the function \( y = f(x) \) given that \( f'(x) = 6x^2 – 4x^{-3} + \dfrac{2}{\sqrt{x}} \) and the curve passes through \( (1, 5) \).
▶️ Answer / Explanation
Integrate term-by-term:
\( \displaystyle \int 6x^2\,dx = 2x^3 \)
\( \displaystyle \int -4x^{-3}\,dx = 2x^{-2} \)
\( \displaystyle \int 2x^{-1/2}\,dx = 4x^{1/2} \)
So,
\( f(x) = 2x^3 + 2x^{-2} + 4x^{1/2} + C \)
Use point \( (1,5) \):
\( 5 = 2(1)^3 + 2(1)^{-2} + 4(1)^{1/2} + C \)
\( 5 = 2 + 2 + 4 + C \)
\( C = -3 \)
Final answer:
\( \boxed{f(x) = 2x^3 + 2x^{-2} + 4\sqrt{x} – 3} \)