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Definition: Integration using Trigonometric Identities
Integration using trigonometric identities means converting a trigonometric expression into an easier form with the help of standard identities before integrating.
Identities
| Identity | Use |
|---|---|
| \(\cos 2x = 2\cos^2 x - 1\) | For \(\cos^2 x\) |
| \(\cos^2 x = \frac{1+\cos 2x}{2}\) | For even power of cosine |
| \(\sin A \cos B = \frac{1}{2}[\sin(A+B)+\sin(A-B)]\) | For product of sine and cosine |
| \(\sin 3x = 3\sin x - 4\sin^3 x\) | For \(\sin^3 x\) |
Example 1
Find
- \[\int \cos^2 x dx\]
- \[\int \sin 2x \cos 3x dx\]
Solution:
(i) Recall the identity \[\cos 2x = 2 \cos^2 x - 1\], which gives
Therefore, \[\int \cos^2 x dx = \frac{1}{2} \int (1 + \cos 2x) dx\]
\[= \frac{x}{2} + \frac{1}{4} \sin 2x + \text{C}\]
(ii) Recall the identity \[\sin x \cos y = \frac{1}{2} [\sin (x + y) + \sin (x - y)]\]
Then \[\int \sin 2x \cos 3x dx = \frac{1}{2} \left[ \int \sin 5x dx - \int \sin x dx \right]\]
\[= \frac{1}{2} \left[ -\frac{1}{5} \cos 5x + \cos x \right] + \text{C}\]
\[= -\frac{1}{10} \cos 5x + \frac{1}{2} \cos x + \text{C}\]
Key Points: Integration Using Trigonometric Identities
-
First inspect the pattern in the integrand.
-
Do not integrate complicated trigonometric expressions directly if an identity can simplify them first.
-
After simplification, integrate term by term carefully.
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Always add the constant of integration, \(C\).
