IB Mathematics SL 5.10 Indefinite integral of xn , sinx, cosx, and ex AA SL Paper 2- Exam Style Questions- New Syllabus
Question
Consider the function \( f(x) = \frac{(x-1)^2}{x} \), defined for all real \( x \) except zero.
(a) Show that \( \frac{(x-1)^2}{x} \) can be rewritten as \( x – 2 + \frac{1}{x} \).
(b) Using your result from part (a), determine \( \displaystyle \int f(x) \, dx \).
Most-appropriate topic codes (Mathematics: analysis and approaches guide):
• Prior learning: Manipulation of algebraic expressions, including factorization and expansion — part (a)
• SL 5.5: Introduction to integration as anti-differentiation of functions of the form \( f(x)=ax^n+bx^{n-1} \) — part (b)
• SL 5.10: Indefinite integral of \( x^n (n \in \mathbb{Q}) \) and \( \frac{1}{x} \) — part (b)
• SL 5.5: Introduction to integration as anti-differentiation of functions of the form \( f(x)=ax^n+bx^{n-1} \) — part (b)
• SL 5.10: Indefinite integral of \( x^n (n \in \mathbb{Q}) \) and \( \frac{1}{x} \) — part (b)
▶️ Answer/Explanation
(a)
Expand the numerator:
\( (x-1)^2 = x^2 – 2x + 1 \)
Divide each term by \( x \):
\( \frac{x^2 – 2x + 1}{x} = \frac{x^2}{x} – \frac{2x}{x} + \frac{1}{x} \)
Simplify the expression:
\( = x – 2 + \frac{1}{x} \)
\( \boxed{x – 2 + \frac{1}{x}} \)
(b)
Substitute the simplified form into the integral:
\( \int f(x) \, dx = \int \left( x – 2 + \frac{1}{x} \right) dx \)
Integrate term-by-term:
\( \int x \, dx = \frac{x^2}{2} \)
\( \int -2 \, dx = -2x \)
\( \int \frac{1}{x} \, dx = \ln|x| \)
Combine the results and add the constant of integration \( C \):
\( \int f(x) \, dx = \frac{x^2}{2} – 2x + \ln|x| + C \)
\( \displaystyle \boxed{\frac{x^2}{2} – 2x + \ln|x| + C} \)