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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 1

Consider the function $f(x)=\sec x-\tan x$ , for $0<x<\frac{\pi}{2}$ .

Show that $f(x)=\frac{1-\sin x}{\cos x}$ .

[3]

Find $f^{\prime}(x)$ , simplifying your answer.

[3]

Hence, find $\int f(x) d x$ .

[4]

Question 6

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 7

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 8

HLPaper 1

Consider the curve defined by $x^{2}+\cos y=1$ , for $0 \leq y \leq \pi$ .

Find $\frac{d y}{d x}$ using implicit differentiation.

[3]

Find the coordinates of the point on the curve where the tangent is horizontal.

[4]

Find the area of the region enclosed by the curve, the x -axis, and the lines $x=0$ and $x=1$ .

[5]

Question 9

HLPaper 2

Consider the identity involving trigonometric functions. (a) Prove that $(\sec x-\tan x)^{2}=\frac{1-\sin x}{1+\sin x}$ . (b) Hence, find $\int_{0}^{\frac{\pi}{4}}(\sec x-\tan x)^{2} d x$ , expressing the answer in the form $a+b \sqrt{2}$ , where $a, b \in \mathbb{Z}$ .

Prove that $(\sec x-\tan x)^{2}=\frac{1-\sin x}{1+\sin x}$ .

[4]

Find $\int_{0}^{\frac{\pi}{4}}(\sec x-\tan x)^{2} d x$ , expressing the answer in the form $a+b \sqrt{2}$ , where $a, b \in \mathbb{Z}$ .

[6]

Question 10

HLPaper 2

In a chemical reaction, the concentration of a reactant is modeled by the curve $y^{2}=$ $\sin (2 x)$ , for $0 \leq x \leq \frac{\pi}{4}$ , where $y$ is the concentration and $x$ is time in seconds.

Use implicit differentiation to find $\frac{d y}{d x}$ .

[4]

Find the equation of the tangent to the curve at $x=\frac{\pi}{6}$ .

[4]

Sketch the curve and the tangent line, indicating intercepts and key points.

[4]

The region bounded by the curve, the x-axis, and $x=\frac{\pi}{4}$ is rotated through $2 \pi$ about the x-axis. Find the volume of the solid generated, giving your answer in terms of $\pi$ .

[6]

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 1

Consider the function $f(x)=\sec x-\tan x$ , for $0<x<\frac{\pi}{2}$ .

Show that $f(x)=\frac{1-\sin x}{\cos x}$ .

[3]

Find $f^{\prime}(x)$ , simplifying your answer.

[3]

Hence, find $\int f(x) d x$ .

[4]

Question 6

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 7

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 8

HLPaper 1

Consider the curve defined by $x^{2}+\cos y=1$ , for $0 \leq y \leq \pi$ .

Find $\frac{d y}{d x}$ using implicit differentiation.

[3]

Find the coordinates of the point on the curve where the tangent is horizontal.

[4]

Find the area of the region enclosed by the curve, the x -axis, and the lines $x=0$ and $x=1$ .

[5]

Question 9

HLPaper 2

Prove that $(\sec x-\tan x)^{2}=\frac{1-\sin x}{1+\sin x}$ .

[4]

Find $\int_{0}^{\frac{\pi}{4}}(\sec x-\tan x)^{2} d x$ , expressing the answer in the form $a+b \sqrt{2}$ , where $a, b \in \mathbb{Z}$ .

[6]

Question 10

HLPaper 2

In a chemical reaction, the concentration of a reactant is modeled by the curve $y^{2}=$ $\sin (2 x)$ , for $0 \leq x \leq \frac{\pi}{4}$ , where $y$ is the concentration and $x$ is time in seconds.

Use implicit differentiation to find $\frac{d y}{d x}$ .

[4]

Find the equation of the tangent to the curve at $x=\frac{\pi}{6}$ .

[4]

Sketch the curve and the tangent line, indicating intercepts and key points.

[4]

The region bounded by the curve, the x-axis, and $x=\frac{\pi}{4}$ is rotated through $2 \pi$ about the x-axis. Find the volume of the solid generated, giving your answer in terms of $\pi$ .

[6]

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 1

Consider the function $f(x)=\sec x-\tan x$ , for $0<x<\frac{\pi}{2}$ .

Show that $f(x)=\frac{1-\sin x}{\cos x}$ .

[3]

Find $f^{\prime}(x)$ , simplifying your answer.

[3]

Hence, find $\int f(x) d x$ .

[4]

Question 6

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 7

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 8

HLPaper 1

Consider the curve defined by $x^{2}+\cos y=1$ , for $0 \leq y \leq \pi$ .

Find $\frac{d y}{d x}$ using implicit differentiation.

[3]

Find the coordinates of the point on the curve where the tangent is horizontal.

[4]

Find the area of the region enclosed by the curve, the x -axis, and the lines $x=0$ and $x=1$ .

[5]

Question 9

HLPaper 2

Prove that $(\sec x-\tan x)^{2}=\frac{1-\sin x}{1+\sin x}$ .

[4]

Find $\int_{0}^{\frac{\pi}{4}}(\sec x-\tan x)^{2} d x$ , expressing the answer in the form $a+b \sqrt{2}$ , where $a, b \in \mathbb{Z}$ .

[6]

Question 10

HLPaper 2

In a chemical reaction, the concentration of a reactant is modeled by the curve $y^{2}=$ $\sin (2 x)$ , for $0 \leq x \leq \frac{\pi}{4}$ , where $y$ is the concentration and $x$ is time in seconds.

Use implicit differentiation to find $\frac{d y}{d x}$ .

[4]

Find the equation of the tangent to the curve at $x=\frac{\pi}{6}$ .

[4]

Sketch the curve and the tangent line, indicating intercepts and key points.

[4]

The region bounded by the curve, the x-axis, and $x=\frac{\pi}{4}$ is rotated through $2 \pi$ about the x-axis. Find the volume of the solid generated, giving your answer in terms of $\pi$ .

[6]

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