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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 integral of 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 $\cos$ and $\sin$

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 $a\ln x+b$ 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 $\int {g\left( y \right)} {\text{ }}dy$ .

[8]

Question 6

HLPaper 1

Using the substitution $v = \sin y$ , find

$\int \frac{{\cos^3 y \, dy}}{{\sqrt {\sin y} }}$ .

[5]

Question 7

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 8

HLPaper 1

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

$\int \frac{dy}{y^{\frac{3}{2}} + y^{\frac{1}{2}}}$ .

[4]

Hence find the value of $\frac{1}{2}\int\limits_1^9 {\frac{{dy}}{{y^{\frac{3}{2}}} + {y^{\frac{1}{2}}}}}$ , expressing your answer in the form arctan $r$ , where $r \in \mathbb{Q}$ .

[3]

Question 9

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 $(h \circ k)(x)$ with range $y\in\R,y\not=\frac{1}{4}$

[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 $\int\! f(x)dx$ .

[5]

Question 10

HLPaper 1

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

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

[7]

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