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AHL 5.16—Integration by substitution, parts and repeated parts - IB Questionbank | RevisionDojo

Practice AHL 5.16—Integration by substitution, parts and repeated parts 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

By using the substitution $u=\sec x$ or otherwise, find an expression for $\int_0^{\frac{\pi}{3}} \sec^n x \tan x, dx$ in terms of $n$ , where $n$ is a non-zero real number.

[6]

Question 2

HLPaper 1

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

[4]

Question 3

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 4

HLPaper 1

Consider the function $f(x) = x^5 \ln(x)$ and its behavior over the interval $[1, e]$ .

Calculate the definite integral $\int_{1}^{e} x^5 \ln(x) \, dx$ .

[4]

Question 5

HLPaper 1

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

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

[5]

Question 6

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 7

HLPaper 1

This question explores integration using substitution and integration by parts.

Evaluate the integral $\int x e^{2x} \, dx$ using integration by parts.

[5]

Evaluate the integral $\int \frac{\ln(x)}{x^2} \, dx$ using integration by parts.

[5]

Evaluate the integral $\int \frac{1}{x \ln(x)} \, dx$ using the substitution $u = \ln(x)$ .

[4]

Evaluate the integral $\int x \sin(x) \, dx$ using integration by parts.

[5]

Evaluate the integral $\int e^{x} \cos(x) \, dx$ using integration by parts twice.

[6]

Question 8

HLPaper 1

Find $\int \arcsin y \,dy$

[5]

Question 9

HLPaper 1

Evaluate: $\int_0^{\pi} \cos(\pi - x) \, dx.$

[3]

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]

AHL 5.16—Integration by substitution, parts and repeated parts Questionbank