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

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

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

[5]

Question 4

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 5

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 6

HLPaper 1

Find $\int \arcsin y \,dy$

[5]

Question 7

HLPaper 1

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

[3]

Question 8

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 9

HLPaper 1

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

Use integration by parts twice to evaluate the integral.

[6]

Sketch the integrand $y=x^{2} e^{x}$ for $0 \leq x \leq 2$ .

[1]

Question 10

HLPaper 2

In a fluid dynamics model, the pressure distribution along a pipe is described by the integral $\int \frac{\cos x}{\sqrt{x}} d x$ , where $x>0$ represents the distance along the pipe.

Use the substitution $x=t^{2}$ followed by integration by parts to find the antiderivative.

[7]

Show that the antiderivative can be expressed in terms of the Fresnel cosine integral $C(z)=\int_{0}^{z} \cos \left(u^{2}\right) d u$ .

[4]

Evaluate $\int_{0}^{1} \frac{\cos x}{\sqrt{x}} d x$ using the result from part (b).

[3]

Sketch the integrand for $0<x \leq 2$ and shade the area from $x=0^{+}$ to $x=1$ . marks]

[3]