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

SL 5.1—Introduction of differential calculus

SL 5.2—Increasing and decreasing functions

SL 5.3—Differentiating polynomials, n E Z

SL 5.4—Tangents and normal

SL 5.5—Integration introduction, areas between curve and x axis

SL 5.6—Differentiating polynomials n E Q. Chain, product and quotient rules

SL 5.7—The second derivative

SL 5.8—Testing for max and min, optimisation. Points of inflexion

SL 5.9—Kinematics problems

SL 5.10—Indefinite integration, reverse chain, by substitution

SL 5.11—Definite integrals, areas under curve onto x-axis and areas between curves

AHL 5.12—First principles, higher derivatives

AHL 5.13—Limits and L’Hopitals

AHL 5.14—Implicit functions, related rates, optimisation

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

AHL 5.16—Integration by substitution, parts and repeated parts

AHL 5.17—Areas under curve onto y-axis, volume of revolution (about x and y axes)

AHL 5.18—1st order DE’s – Euler method, variables separable, integrating factor, homogeneous DE using sub y=vx

AHL 5.19—Maclaurin series

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

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 $\int {g\left( y \right)} {\text{ }}dy$ .

[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 $\int_0^2\! g(x)dx$

[5]

Question 4

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 $A, B, C \in \mathbb{R}$ .

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

[2]

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

[5]

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

[3]

Question 5

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.

[2]

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

[3]

Question 6

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.

[4]

Use parts (a) and (b) to show that ${\text{ln}}\left( 1 + t \right) - {\text{ln}}\,t < {\text{ln}}\,2$ .

[1]

Question 7

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 ∈ ℚ. Find the value of a and the value of b.

[4]

Find the coordinates where the graph of g crosses the x-axis;

[1]

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

[3]

Question 8

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]

Question 9

HLPaper 1

Consider the identity $\frac{4x^2 + 7x - 1}{(x-1)(x+1)(x-2)} \equiv \frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{x-2}$ , where $A, B, C \in \mathbb{R}$ .

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

[5]

Hence, express $\frac{4x^2 + 7x - 1}{(x-1)(x+1)(x-2)}$ as the sum of partial fractions.

[2]

Determine the value of the integral $\int \frac{4x^2 + 7x - 1}{(x-1)(x+1)(x-2)} \, dx$ .

[3]

Paper

Level

Question 1

HLPaper 1

Let $g(y) = \frac{1}{{1 - {y^2}}}$ for $- 1 < y < 1$ . Use partial fractions to find $\int {g\left( y \right)} {\text{ }}dy$ .

[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 $\int_0^2\! g(x)dx$

[5]

Question 4

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 $A, B, C \in \mathbb{R}$ .

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

[2]

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

[5]

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

[3]

Question 5

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.

[2]

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

[3]

Question 6

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.

[4]

Use parts (a) and (b) to show that ${\text{ln}}\left( 1 + t \right) - {\text{ln}}\,t < {\text{ln}}\,2$ .

[1]

Question 7

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 ∈ ℚ. Find the value of a and the value of b.

[4]

Find the coordinates where the graph of g crosses the x-axis;

[1]

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

[3]

Question 8

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]

Question 9

HLPaper 1

Consider the identity $\frac{4x^2 + 7x - 1}{(x-1)(x+1)(x-2)} \equiv \frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{x-2}$ , where $A, B, C \in \mathbb{R}$ .

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

[5]

Hence, express $\frac{4x^2 + 7x - 1}{(x-1)(x+1)(x-2)}$ as the sum of partial fractions.

[2]

Determine the value of the integral $\int \frac{4x^2 + 7x - 1}{(x-1)(x+1)(x-2)} \, dx$ .

[3]