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AHL 5.15—Further derivatives and indefinite integration of these, partial fractions - IB Questionbank | RevisionDojo

Practice AHL 5.15—Further derivatives and indefinite integration of these, partial fractions with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Let $g(y) = \frac{1}{1 - y^2}$ for $-1 < y < 1$ . Use partial fractions to find $\int g(y) \, \mathrm{d}y$ .

[8]

Question 2

HLPaper 1

Consider the rational function $f(x) = \frac{2x^2 + 3x + 1}{(x+1)^2(x-2)}$ .

Express $f(x)$ as a sum of partial fractions.

[7]

Hence, or otherwise, find $\int f(x) \, dx$ with only one $\ln$ term.

[4]

Question 3

HLPaper 1

Consider the rational function $g(x) = \frac{3x^2 + 5x + 2}{(x-1)^2(x+2)}$ .

Express $g(x)$ as a sum of partial fractions.

[8]

Hence, or otherwise, compute $\int_2^3 g(x) \, dx$ .

[5]

Question 4

HLPaper 1

Evaluate the following definite integral: $\int_0^1 \frac{x}{\sqrt{4 - x^4}} \, dx.$

$\int_0^1 \frac{x}{\sqrt{4 - x^4}} \, dx.$

[6]

Question 5

HLPaper 2

Consider the identity $\frac{3x^2 + 5x - 2}{(x-1)(x+2)(x-3)} \equiv \frac{A}{x-1} + \frac{B}{x+2} + \frac{C}{x-3}$ , where $A, B, C \in \mathbb{R}$ .

Find the values of $A$ , $B$ , and $C$ .

[5]

Hence, express $\frac{3x^2 + 5x - 2}{(x-1)(x+2)(x-3)}$ as the sum of partial fractions.

[2]

Determine the value of the integral $\int \frac{3x^2 + 5x - 2}{(x-1)(x+2)(x-3)} \, dx$ .

[3]

Question 6

HLPaper 2

Consider the identity $\frac{2x^2 + 3x - 5}{(x+1)(x-2)(x-3)} \equiv \frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{x-3}$ , where $A, B, C \in \mathbb{R}$ .

Find the values of $A$ , $B$ and $C$ .

[5]

Hence, express $\frac{2x^2 + 3x - 5}{(x+1)(x-2)(x-3)}$ as the sum of partial fractions.

[1]

Find the integral $\int \frac{2x^2 + 3x - 5}{(x+1)(x-2)(x-3)} \, dx$ .

[5]

Question 7

HLPaper 1

Let $f(x) = \dfrac{4x^2 + 2x - 1}{x(2x-1)(x+1)}$ for $x > 1$ .

Express $f(x)$ in partial fractions of the form $\dfrac{A}{x} + \dfrac{B}{2x-1} + \dfrac{C}{x+1}$ .

[5]

Hence, find the exact area of the region bounded by the curve $y=f(x)$ , the $x$ -axis, and the lines $x=2$ and $x=3$ .

[6]

Question 8

HLPaper 1

Consider the integral $\int\limits_1^t \frac{-1}{y + y^2} dy$ for $t > 1$ .

Express the function $g(y) = \frac{-1}{{y + y^2}}$ in partial fractions.

[6]

Very briefly, explain why the value of this integral must be negative.

[1]

Use parts 1 and 2 to show that $\ln(1 + t) - \ln t < \ln 2$ .

[4]

Question 9

HLPaper 2

Consider the function $g(x) = \frac{x^2 - x - 12}{2x - 15}, x \in \mathbb{R}, x \neq \frac{15}{2}$ .

Find the coordinates where the graph of $g$ crosses the $y$ -axis.

[2]

Write down the equation of the vertical asymptote of the graph of $g$ .

[1]

Express $\frac{1}{g(x)}$ in partial fractions.

[3]

Hence find the exact value of $\int_{0}^{3} \frac{1}{g(x)} dx$ , expressing your answer as a single logarithm.

[4]

The oblique asymptote of the graph of $g$ can be written as $y = ax + b$ where $a, b \in \mathbb{Q}$ . Find the value of $a$ and the value of $b$ .

[4]

Find the coordinates where the graph of $g$ crosses the $x$ -axis.

[2]

Sketch the graph of $g$ for $-30 \leq x \leq 30$ , clearly indicating the points of intersection with each axis and any asymptotes.

[3]

Question 10

HLPaper 2

Consider the expression $\frac{3x^2 + 5x - 2}{(x-1)(x^2 + 3x + 2)}$

Express $\frac{3x^2 + 5x - 2}{(x-1)(x^2 + 3x + 2)}$ in partial fractions.

[8]

Integrate the partial fractions found in part 1.

[4]

AHL 5.15—Further derivatives and indefinite integration of these, partial fractions Questionbank