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SL 5.10—Indefinite integration, reverse chain, by substitution - IB Questionbank | RevisionDojo

Practice SL 5.10—Indefinite integration, reverse chain, by substitution 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

Find the integral $\int \frac{x}{\sqrt{1-x^2}} \, dx$ using the substitution $u = 1 - x^2$ .

[4]

Question 2

SLPaper 1

Consider the function $f(x) = \frac{2x}{x^2 + 1}$ .

Evaluate the integral $\int \frac{2x}{x^2 + 1} \, dx$ .

[5]

Question 3

SLPaper 1

Let $g'(x) = \frac{8x}{\sqrt{2x^2 + 1}}$ . Given that $g(0) = 5$ , find $g(x)$ .

[5]

Question 4

HLPaper 1

Consider the double angle identities of cosine and sine.

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

[4]

Hence, find $\int(\tan(2x)-\tan(2x)\tan^2(x))\,dx$ in the form $a\ln|\sec(x)|+C$ where $a>0$ .

[6]

Question 5

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 6

SLPaper 1

The following question involves indefinite integration using the reverse chain rule by substitution.

Evaluate the integral $\int \frac{2x}{(x^2 + 3)^2} \, dx$ .

[5]

Question 7

HLPaper 1

Use the substitution $v = y^{\frac{1}{2}}$ to find

$\int \frac{\mathrm{d}y}{y^{\frac{3}{2}} + y^{\frac{1}{2}}}$

[4]

Hence find the value of $\frac{1}{2}\int_{1}^{9} \frac{\mathrm{d}y}{y^{\frac{3}{2}} + y^{\frac{1}{2}}}$ , expressing your answer in the form $\arctan r$ , where $r \in \mathbb{Q}$ .

[3]

Question 8

HLPaper 1

Let $h(x) = \frac{1}{x^2}$ and $k(x) = \ln(x)$ .

Find the composite function $(h \circ k)(x)$ .

[1]

Determine the domain of the composite function $(h \circ k)(x)$ , given that $(h \circ k)(x) \neq \frac{1}{4}$ .

[5]

Evaluate the limit $\lim_{x \to 1} (h \circ k)(x)$ .

[2]

Now consider the function $f(x)=k(x)\sqrt{h(x)}$ . Hence, find $\int f(x) \, \mathrm{d}x$ .

[5]

Question 9

HLPaper 1

Consider the integral of the function $f(x) = (4x^3 - 9x^2 + 7x + 3)e^{-x}$ with respect to $x$ .

Evaluate the integral $\int (4x^3 - 9x^2 + 7x + 3)e^{-x} \, dx$

[7]

Question 10

SLPaper 2

The derivative of a function $h$ is given by $h'(x) = 3x^2 + 5e^x$ , where $x \in \mathbb{R}$ . The graph of $h$ passes through the point $(0, 4)$ . Find $h(x)$ .

[5]

SL 5.10—Indefinite integration, reverse chain, by substitution Questionbank