IB Mathematics Analysis and Approaches Topic SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 5.10—indefinite Integration, Reverse Chain, by Substitution with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 5.10—indefinite Integration, Reverse Chain, by Substitution and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

A function $g (x)$ is defined for $x > 0$ with derivative $g^{'} (x) = \frac{4}{x x}$ . The region enclosed by the graph of $y = g (x)$ , the x-axis, the line $x = 1$ , and the line $x = 4$ is rotated $36 0^{\circ}$ about the x-axis. The resulting volume is $64 π (ln 4 - 1)$ cubic units.

Find $\int \frac{4}{x x} d x$ .

[4]

Find $g (x)$ , given that $g (1) = 0$ .

[3]

Show that the volume of the solid is $64 π (ln 4 - 1)$ cubic units.

[4]

Question 2

SLPaper 1

A function $h (x)$ for $x \geq 0$ has derivative $h^{'} (x) = \frac{8 x}{( 2 x ^{2} + 1 ) ^{2}}$ .

Show that $\int \frac{8 x}{( 2 x ^{2} + 1 ) ^{2}} d x = - \frac{2}{2 x ^{2} + 1} + c$ .

[4]

Given that $h (0) = 3$ , find $h (x)$ .

[3]

Find the gradient of the tangent to the graph of $h$ at $x = 1$ .

[2]

Question 3

HLPaper 1

A function $g (x)$ for $x > 0$ has derivative $g^{'} (x) = \frac{6 x ^{2}}{( x ^{3} + 1 ) ^{2}}$ . The graph of $g$ intersects the line $y = x$ at points $A (1, 1)$ and $B$ . The region enclosed by the graph of $g$ , the line $y = x$ , and the lines $x = 1$ and $x = k$ (where $k > 1$ ) has an area of $\frac{23}{18}$ .

Show that $\int \frac{6 x ^{2}}{( x ^{3} + 1 ) ^{2}} d x = - \frac{2}{x ^{3} + 1} + c$ .

[5]

Find $g (x)$ , given that $g (1) = 1$ .

[4]

Find the coordinates of point $B$ .

[5]

The region enclosed by the graph of $g^{'}$ , the $x$ -axis, and the lines $x = 1$ and $x = k$ (where $k > 1$ ) has an area of $\frac{7}{9}$ . Find the value of $k$ .

[6]

Question 4

HLPaper 1

Let $f (x) = \frac{l n ( x )}{x}$ for $x > 0$ .

Find the area of the region bounded by the graph of $f$ , the x-axis, and the lines $x = 1$ and $x = e^{2}$ .

[5]

Find the coordinates of the maximum point of the graph of $f$ .

[5]

Show that the x-coordinate of the point of inflection is $e^{3/2}$ .

[5]

The region $R$ , bounded by the graph of $f$ , the x-axis, and the lines $x = 1$ and $x = e^{2}$ , is rotated $36 0^{\circ}$ about the x-axis to form a solid of revolution. Find the volume of this solid.

[6]

Let $S$ be the region bounded by the graph of $f$ , the line tangent to the graph at $x = 1$ , and the line $x = e$ . Find the area of $S$ .

[6]

Question 5

SLPaper 1

Consider a function $f (x)$ for $x > 0$ . The derivative is given by $f^{'} (x) = \frac{2 x}{x ^{2} + 1}$ .

Show that $\int \frac{2 x}{x ^{2} + 1} d x = ln (x^{2} + 1) + c$ .

[3]

Given that $f (1) = 2$ , find $f (x)$ .

[4]

IB Mathematics Analysis and Approaches SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank

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IB Mathematics Analysis and Approaches SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank

Unlock all 43 questions with a free account