IB Mathematics Applications & Interpretation Topic AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 5.11—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 AHL 5.11—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 $f (x)$ has derivative $f^{'} (x) = \frac{6 x ^{2}}{x ^{3} + 8}$ , for $x > 0$ . The graph of $f$ passes through the point $(2, 5)$ .

Find $\int \frac{6 x ^{2}}{x ^{3} + 8} d x$ .

[4]

Hence, find $f (x)$ .

[3]

For $x > 0$ , determine the $x$ -coordinate where the tangent to $f$ is horizontal.

[2]

Question 2

HLPaper 2

Let $y = cos^{6} (3 x)$ .

Consider the function $h (x) = 4 - 3 sin (3 x) cos^{5} (3 x)$ , defined for $0 \leq x \leq \frac{π}{6}$ .

The region $R$ is bounded by the graph of $h$ , the $x$ -axis, the $y$ -axis, and the line $x = \frac{π}{6}$ .

Determine the derivative $\frac{d y}{d x}$ .

[3]

Using your result from the previous part, find $\int sin (3 x) cos^{5} (3 x) d x$ .

[3]

State a definite integral that represents the area of $R$ .

[2]

Determine the precise area of $R$ .

[6]

Question 3

HLPaper 1

Given that $\int_{- 2}^{2} g (y) d y = 10$ and $\int_{0}^{2} g (y) d y = 12$ , find

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

[4]

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

[2]

Question 4

HLPaper 2

Let $y = sin^{5} (2 x)$ .

Consider the function $g (x) = 3 - 2 sin^{4} (2 x) cos (2 x)$ , defined for $0 \leq x \leq \frac{π}{4}$ .

The region $S$ is bounded by the graph of $g$ , the $x$ -axis, the $y$ -axis, and the line $x = \frac{π}{4}$ .

Determine the derivative $\frac{d y}{d x}$ .

[3]

Using your result from the previous part, find $\int sin^{4} (2 x) cos (2 x) d x$ .

[3]

State a definite integral that represents the area of $S$ .

[2]

Determine the precise area of $S$ .

[6]

Question 5

HLPaper 1

Find $\int x e^{1 - x^{2}} d x$ .

[4]

Find $f (x)$ , given that $f^{'} (x) = x e^{1 - x^{2}}$ and $f (1) = 2$ .

[3]

IB Mathematics Applications & Interpretation AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank

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IB Mathematics Applications & Interpretation AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Questionbank

Unlock all 34 questions with a free account