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AHL 5.11—Indefinite integration, reverse chain, by substitution - IB Questionbank | RevisionDojo

Practice AHL 5.11—Indefinite integration, reverse chain, by substitution with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Given that $\int_{-2}^2 g(y) \, \mathrm{d}y = 10$ and $\int_0^2 g(y) \, \mathrm{d}y = 12$ , find

$\int_{-2}^0 (g(y) + 2) \, \mathrm{d}y$ .

[4]

$\int_{-2}^0 g(y+2) \, \mathrm{d}y$ .

[2]

Question 2

HLPaper 1

In a study of renewable energy, the variation of solar energy output can be modeled by the function $y = \sin(x)$ , where $x$ represents the time from sunrise ( $x=0$ ) to sunset ( $x=\pi$ ).

Find the total energy produced from sunrise to sunset, represented by the area between the curve $y = \sin(x)$ and the $x$ -axis.

[5]

Question 3

HLPaper 1

A function $k(x)$ has derivative $k'(x) = \frac{3 \cos x}{\sin^2 x + 1}$ . The graph of $k$ passes through the point $\left(\frac{\pi}{2}, 0\right)$ .

Find $k(x)$ .

[5]

Determine the $x$ -coordinate of the point where the graph of $k$ has a horizontal tangent in the interval $0 \leq x \leq \pi$ .

[2]

Question 4

SLPaper 2

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a = 3t^2 - 14t + 8$, for $0 \le t \le 5$.

When $t = 0$, the velocity of P is $3 \text{ m s}^{-1}$.

Write down the values of $t$ when $a = 0$.

[2]

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

[2]

Find an expression for the velocity of P at time $t$.

[6]

Find the total distance travelled by P when its velocity is increasing.

[4]

Question 5

HLPaper 1

In a renewable energy project, engineers are analyzing the efficiency of solar panels over a specific area. They need to calculate various integrals to determine the total energy produced by the panels over time and space.

Calculate the indefinite integral of the energy output function: $\int E(x) \, dx = \int (3x^2 + 5x + 4) \, dx$

[3]

Evaluate the definite integral: $\int_0^2 (2x + 1) \, dx$

[4]

Use substitution to find the integral: $\int \sin(2x + 5) \, dx$

[5]

Consider the function $f(x) = 4x - x^2$ . Find the area under the curve of $f(x)$ from $x = 0$ to $x = 2$ .

[6]

Question 6

HLPaper 1

In the context of environmental science, we often analyze data related to population growth and resource consumption. A researcher is studying the relationship between the amount of a pollutant in a river and the distance from its source. The concentration of the pollutant can be modeled by the function $\frac{1}{3x + 2}$ , where $x$ represents the distance in kilometers from the source.

Use substitution to find the indefinite integral: $\int \frac{1}{3x + 2} \, dx$

[4]

Question 7

HLPaper 2

A particle $P$ moves along the $x$ -axis. The velocity of $P$ is $v \, \text{m s}^{-1}$ at time $t$ seconds, where $v = -2t^2 + 16t - 24$ for $t \geq 0$ .

Find the times when $P$ is at instantaneous rest.

[2]

Find the magnitude of the particle's acceleration at 6 seconds.

[3]

Find the greatest speed of $P$ in the interval $0 \leq t \leq 6$ .

[2]

The particle starts from the origin $O$ . Find an expression for the displacement of $P$ from $O$ at time $t$ seconds.

[4]

Find the total distance travelled by $P$ in the interval $0 \leq t \leq 4$ .

[3]

Question 8

HLPaper 1

The sides of a bowl are formed by rotating the curve $y = 6 \ln x, 0 \le y \le 9$ , about the y-axis, where x and y are measured in centimetres. The bowl contains water to a height of $h$ cm.

Show that the volume of water, $V$ , in terms of $h$ is $V = 3\pi(e^{h/3} - 1)$ .

[5]

Hence find the maximum capacity of the bowl in $\mathrm{cm}^3$ .

[2]

Question 9

HLPaper 1

A function $h(x)$ has derivative $h'(x) = \frac{8x}{(x^2 + 4)^2}$ . The graph of $h$ passes through the point $(0, 1)$ .

Show that $h(x) = -\frac{4}{x^2 + 4} + c$ .

[4]

Find $h(x)$ .

[2]

Find the area of the region bounded by the graph of $h'(x)$ , the $x$ -axis, and the lines $x = 0$ and $x = 2$ .

[3]

Question 10

HLPaper 2

Let $f'(x) = \frac{\cos x}{\sin^2 x + 4}$ . The region $R$ is enclosed by the graph of $f'$ , the $x$ -axis, and the lines $x = 0$ and $x = \pi$ .

Find $\int \frac{\cos x}{\sin^2 x + 4} \mathrm{d}x$ .

[4]

Find the area of $R$ .

[3]

Sketch the graph of $f'(x)$ for $0 \leq x \leq \pi$ .

[2]

Paper

Level

Question 1

HLPaper 1

Given that $\int_{-2}^2 g(y) \, \mathrm{d}y = 10$ and $\int_0^2 g(y) \, \mathrm{d}y = 12$ , find

$\int_{-2}^0 (g(y) + 2) \, \mathrm{d}y$ .

[4]

$\int_{-2}^0 g(y+2) \, \mathrm{d}y$ .

[2]

Question 2

HLPaper 1

In a study of renewable energy, the variation of solar energy output can be modeled by the function $y = \sin(x)$ , where $x$ represents the time from sunrise ( $x=0$ ) to sunset ( $x=\pi$ ).

Find the total energy produced from sunrise to sunset, represented by the area between the curve $y = \sin(x)$ and the $x$ -axis.

[5]

Question 3

HLPaper 1

A function $k(x)$ has derivative $k'(x) = \frac{3 \cos x}{\sin^2 x + 1}$ . The graph of $k$ passes through the point $\left(\frac{\pi}{2}, 0\right)$ .

Find $k(x)$ .

[5]

Determine the $x$ -coordinate of the point where the graph of $k$ has a horizontal tangent in the interval $0 \leq x \leq \pi$ .

[2]

Question 4

SLPaper 2

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a = 3t^2 - 14t + 8$, for $0 \le t \le 5$.

When $t = 0$, the velocity of P is $3 \text{ m s}^{-1}$.

Write down the values of $t$ when $a = 0$.

[2]

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

[2]

Find an expression for the velocity of P at time $t$.

[6]

Find the total distance travelled by P when its velocity is increasing.

[4]

Question 5

HLPaper 1

Calculate the indefinite integral of the energy output function: $\int E(x) \, dx = \int (3x^2 + 5x + 4) \, dx$

[3]

Evaluate the definite integral: $\int_0^2 (2x + 1) \, dx$

[4]

Use substitution to find the integral: $\int \sin(2x + 5) \, dx$

[5]

Consider the function $f(x) = 4x - x^2$ . Find the area under the curve of $f(x)$ from $x = 0$ to $x = 2$ .

[6]

Question 6

HLPaper 1

Use substitution to find the indefinite integral: $\int \frac{1}{3x + 2} \, dx$

[4]

Question 7

HLPaper 2

A particle $P$ moves along the $x$ -axis. The velocity of $P$ is $v \, \text{m s}^{-1}$ at time $t$ seconds, where $v = -2t^2 + 16t - 24$ for $t \geq 0$ .

Find the times when $P$ is at instantaneous rest.

[2]

Find the magnitude of the particle's acceleration at 6 seconds.

[3]

Find the greatest speed of $P$ in the interval $0 \leq t \leq 6$ .

[2]

The particle starts from the origin $O$ . Find an expression for the displacement of $P$ from $O$ at time $t$ seconds.

[4]

Find the total distance travelled by $P$ in the interval $0 \leq t \leq 4$ .

[3]

Question 8

HLPaper 1

The sides of a bowl are formed by rotating the curve $y = 6 \ln x, 0 \le y \le 9$ , about the y-axis, where x and y are measured in centimetres. The bowl contains water to a height of $h$ cm.

Show that the volume of water, $V$ , in terms of $h$ is $V = 3\pi(e^{h/3} - 1)$ .

[5]

Hence find the maximum capacity of the bowl in $\mathrm{cm}^3$ .

[2]

Question 9

HLPaper 1

A function $h(x)$ has derivative $h'(x) = \frac{8x}{(x^2 + 4)^2}$ . The graph of $h$ passes through the point $(0, 1)$ .

Show that $h(x) = -\frac{4}{x^2 + 4} + c$ .

[4]

Find $h(x)$ .

[2]

Find the area of the region bounded by the graph of $h'(x)$ , the $x$ -axis, and the lines $x = 0$ and $x = 2$ .

[3]

Question 10

HLPaper 2

Let $f'(x) = \frac{\cos x}{\sin^2 x + 4}$ . The region $R$ is enclosed by the graph of $f'$ , the $x$ -axis, and the lines $x = 0$ and $x = \pi$ .

Find $\int \frac{\cos x}{\sin^2 x + 4} \mathrm{d}x$ .

[4]

Find the area of $R$ .

[3]

Sketch the graph of $f'(x)$ for $0 \leq x \leq \pi$ .

[2]

Paper

Level

Question 1

HLPaper 1

Given that $\int_{-2}^2 g(y) \, \mathrm{d}y = 10$ and $\int_0^2 g(y) \, \mathrm{d}y = 12$ , find

$\int_{-2}^0 (g(y) + 2) \, \mathrm{d}y$ .

[4]

$\int_{-2}^0 g(y+2) \, \mathrm{d}y$ .

[2]

Question 2

HLPaper 1

In a study of renewable energy, the variation of solar energy output can be modeled by the function $y = \sin(x)$ , where $x$ represents the time from sunrise ( $x=0$ ) to sunset ( $x=\pi$ ).

Find the total energy produced from sunrise to sunset, represented by the area between the curve $y = \sin(x)$ and the $x$ -axis.

[5]

Question 3

HLPaper 1

A function $k(x)$ has derivative $k'(x) = \frac{3 \cos x}{\sin^2 x + 1}$ . The graph of $k$ passes through the point $\left(\frac{\pi}{2}, 0\right)$ .

Find $k(x)$ .

[5]

Determine the $x$ -coordinate of the point where the graph of $k$ has a horizontal tangent in the interval $0 \leq x \leq \pi$ .

[2]

Question 4

SLPaper 2

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a = 3t^2 - 14t + 8$, for $0 \le t \le 5$.

When $t = 0$, the velocity of P is $3 \text{ m s}^{-1}$.

Write down the values of $t$ when $a = 0$.

[2]

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

[2]

Find an expression for the velocity of P at time $t$.

[6]

Find the total distance travelled by P when its velocity is increasing.

[4]

Question 5

HLPaper 1

Calculate the indefinite integral of the energy output function: $\int E(x) \, dx = \int (3x^2 + 5x + 4) \, dx$

[3]

Evaluate the definite integral: $\int_0^2 (2x + 1) \, dx$

[4]

Use substitution to find the integral: $\int \sin(2x + 5) \, dx$

[5]

Consider the function $f(x) = 4x - x^2$ . Find the area under the curve of $f(x)$ from $x = 0$ to $x = 2$ .

[6]

Question 6

HLPaper 1

Use substitution to find the indefinite integral: $\int \frac{1}{3x + 2} \, dx$

[4]

Question 7

HLPaper 2

A particle $P$ moves along the $x$ -axis. The velocity of $P$ is $v \, \text{m s}^{-1}$ at time $t$ seconds, where $v = -2t^2 + 16t - 24$ for $t \geq 0$ .

Find the times when $P$ is at instantaneous rest.

[2]

Find the magnitude of the particle's acceleration at 6 seconds.

[3]

Find the greatest speed of $P$ in the interval $0 \leq t \leq 6$ .

[2]

The particle starts from the origin $O$ . Find an expression for the displacement of $P$ from $O$ at time $t$ seconds.

[4]

Find the total distance travelled by $P$ in the interval $0 \leq t \leq 4$ .

[3]

Question 8

HLPaper 1

The sides of a bowl are formed by rotating the curve $y = 6 \ln x, 0 \le y \le 9$ , about the y-axis, where x and y are measured in centimetres. The bowl contains water to a height of $h$ cm.

Show that the volume of water, $V$ , in terms of $h$ is $V = 3\pi(e^{h/3} - 1)$ .

[5]

Hence find the maximum capacity of the bowl in $\mathrm{cm}^3$ .

[2]

Question 9

HLPaper 1

A function $h(x)$ has derivative $h'(x) = \frac{8x}{(x^2 + 4)^2}$ . The graph of $h$ passes through the point $(0, 1)$ .

Show that $h(x) = -\frac{4}{x^2 + 4} + c$ .

[4]

Find $h(x)$ .

[2]

Find the area of the region bounded by the graph of $h'(x)$ , the $x$ -axis, and the lines $x = 0$ and $x = 2$ .

[3]

Question 10

HLPaper 2

Let $f'(x) = \frac{\cos x}{\sin^2 x + 4}$ . The region $R$ is enclosed by the graph of $f'$ , the $x$ -axis, and the lines $x = 0$ and $x = \pi$ .

Find $\int \frac{\cos x}{\sin^2 x + 4} \mathrm{d}x$ .

[4]

Find the area of $R$ .

[3]

Sketch the graph of $f'(x)$ for $0 \leq x \leq \pi$ .

[2]

AHL 5.11—Indefinite integration, reverse chain, by substitution Questionbank