FAD1014: MATHEMATICS II — Tutorial 4

Centre for Foundation Studies in Science, University of Malaya
Session 2025/2026

Question 1: Trigonometric Substitutions

Using appropriate trigonometric substitutions, find:

(a) $\int \frac{x^2}{\sqrt{x^2 + 4}} dx$

(b) $\int \frac{dx}{x^2\sqrt{4 - x^2}}$

(c) $\int \frac{x^3}{\sqrt{16 - x^2}} dx$

(d) $\int \frac{dx}{x\sqrt{x^2 - 25}}$

(e) $\int \frac{dx}{(4x - 9)^{3/2}}$

(f) $\int \frac{dx}{x^2\sqrt{x^2 + 9}}$

Hint: $\int \csc \alpha , d\alpha = -\ln|\csc \alpha + \cot \alpha| + c$

Question 2: Trigonometric Substitution Application

If $\int \sec \alpha , d\alpha = \ln|\sec \alpha + \tan \alpha| + c$, find $\int \frac{1}{\sqrt{2x^2 - 4}} dx$ using suitable trigonometric substitution.

Question 3: Definite Integral with Absolute Value

Evaluate the definite integral:

$$\int_0^4 |x^2 - 2| dx$$

Question 4: Partial Fractions

Evaluate the following integrations by simplifying the rational functions using partial fractions:

(a) $\int \frac{dx}{(x + 4)(x - 1)}$

(b) $\int \frac{x + 5}{x^2 - 1} dx$

Question 5: Inverse Tangent Integrals

Given that $\int \frac{1}{x^2 + 1} dx = \tan^{-1} x + C$, find the following integrations:

(a) $\int \frac{x^3 - 2x^2 + 3}{x^4 + x^2} dx$

(b) $\int \frac{5x^3 - 4x^2 + 2x - 3}{x^4 + x^2} dx$

Question 6: Definite Integrals

Evaluate the following definite integrals:

(a) $\int_1^2 (x^2 - x)(x + \frac{1}{x}) dx$

(b) $\int_0^1 (1 - x)\sqrt{x} , dx$

(c) $\int_1^2 (2 - 3x)^2 dx$

(d) $\int_0^{\pi/4} (\sin x + \cos x) dx$

(e) $\int_0^1 (e^x + 5^x) dx$

(f) $\int_1^e (x - \frac{1}{x}) dx$

(g) $\int_0^{\pi/4} \sin^2 2\theta , d\theta$

(h) $\int_0^4 |x - 2| dx$

Related Concepts

Related Lectures

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