FAD1014: MATHEMATICS II — Tutorial 3

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

Question 1: Evaluate the Following Integrals

(a) $\int \tan^7 x , dx$

(b) $\int \cos^5 x , dx$

(c) $\int \sin^3 2x \cos^4 2x , dx$

(d) $\int \tan^5 x \sec^2 x , dx$

Show that:

$$\int \cos^4 x \sin^2 x , dx = \frac{1}{16}\left(x - \frac{\sin 4x}{4} + \frac{\sin^3 2x}{3}\right) + c$$

Find:

(a) $\int \sin^2 x , dx$

(b) $\int \cos ax , dx$

(c) $\int \cos^4 2x , dx$

(d) $\int \tan 3x , dx$

Evaluate the following trigonometric integrals using appropriate techniques:

(a) $\int \sin^2 x \cos^3 x , dx$

(b) $\int \sin^3 x \cos^3 x , dx$

(c) $\int \sin^5 x \cos^3 x , dx$

(d) $\int \cos^2 x \sin^2 x , dx$

Find the following integrals:

(a) $\int \sin 2x \cos 4x , dx$

(b) $\int \cos 2x \cos 4x , dx$

(c) $\int \sin 2x \sin 4x , dx$

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