Use trigonometrical relationships to facilitate complex trigonometric integrals.
Integrate by decomposing into partial fractions.
Integration by u-Substitution
Make x equal to something; when differentiated, multiply the substituted form directly.
Make u equal to something; when differentiated, multiply the substituted form with its reciprocal.
With definite integrals, change the limits in terms of u.
Examples
The diagram shows part of curve y = sin3 2x cos3 2x. The shaded region shown is bounded by the curve and the x-axis, and its exact area is denoted by A. Use the substitution u = sin 2x in a suitable integral to find the value of A.
To find the limit, you are trying to find the points at which y = 0;
sin x = 0 at x = 0, π, 2π
cos x = 0 at x = π/2, 3π/4
Choose the two closest to 0 because the shaded area has gone through y = 0 only twice, therefore 0 and π/2
Since it is sin 2x and cos 2x, divide both limits by 2, therefore, limits are 0 and π/4
Integrate by u-substitution, let:
u = sin 2x
du/dx = 2 cos 2x
dx/du = 1/2 cos 2x
sin3 2x cos3 2x ≡ (sin 2x)3 (cos 2x)2 cos 2x
≡ (sin3 2x × (1 – sin2 2x)) cos 2x
≡ (sin3 2x – sin5 2x) cos 2x
f(x) dx/du = (sin3 2x – sin5 2x) cos 2x × 1/2 cos 2x