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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

A function $k(x)$ has derivative $k^{\prime}(x)=\frac{3 \cos x}{\sin ^{2} x+1}$ . 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 2

HLPaper 1

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

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 $h^{\prime}(x)$ , the x -axis, and the lines $x=0$ and $x=2$ .

[3]

Question 3

HLPaper 2

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

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

[4]

Find the area of $R$ .

[3]

Sketch the graph of $f^{\prime}(x)$ for $0 \leq x \leq \pi$ .

[2]

Question 4

HLPaper 1

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

$\int_{ - 2}^0 {\left( {g\left( y \right){\text{ + 2}}} \right){\text{d}}y}$ .

$\int_{ - 2}^0 {g\left( {y{\text{ + 2}}} \right){\text{d}}y}$ .

Question 5

HLPaper 1

In a study of renewable energy, the oscillation of solar energy output can be modeled by the function $y = \cos(x)$ , where $x$ is measured in hours from sunrise to sunset.

Find the area between the curve $y = \cos(x)$ and the $x$ -axis from $x = 0$ to $x = \pi$ .

[5]

Question 6

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 $a = 3{t^2} - 14t + 8$ , for $0 \leqslant t \leqslant 5$ .

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

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 7

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]

Find the area under the curve of $f(x)$ from $x = 0$ to $x = 2$ : $\int_0^2 (4x - x^2) \, dx$

[6]

Question 8

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 solve the integral that represents the total concentration of the pollutant over a certain distance: $\int \frac{1}{3x + 2} \, dx$

[4]

Question 9

HLPaper 2

A particle P moves along the x-axis. The velocity of P is v ms^(-1) at time t seconds, where v = -2t^2 + 16t - 24 for t ≥ 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 ≤ t ≤ 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 ≤ t ≤ 4.

[3]

Question 10

HLPaper 1

The sides of a bowl are formed by rotating the curve $y = 6 \ln x, 0 ≤ y ≤ 9$ , 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 $V = 3π(e^{(h/3)} - 1)$ .

[5]

Hence find the maximum capacity of the bowl in $cm^3$ .

[2]

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