Evaluate the integral. ∫cos23xdx

Question

Evaluate the integral.  ∫cos23xdx

Bella · Accepted Answer

Step 1
The angles and the sides of a triangle are related by the trigonometric ratios. Trigonometric ratios are the ratios of the sides of the triangle.
The cosine of an angle is the ratio of the base and the hypotenuse of the triangle. The cosine function can be expressed in the exponential form by the expression

\cos x = \frac{e^{i x} + e^{- i x}}{2}

.
Step 2
The cosine function in the given integral is expressed in the exponential form by the expression

\cos x = \frac{e^{i x} + e^{- i x}}{2}

as follows:

\int \cos^{2} 3 x d x = \int {(\frac{e^{3 i x} + e^{- 3 i x}}{2})}^{2} d x

= \int (\frac{e^{6 i x} + e^{- 6 i x} + 2}{4}) d x

= \frac{1}{4} \int (e^{6 i x} + e^{- 6 i x} + 2) d x

= \frac{1}{4} (\frac{e^{6 i x}}{6 i} - \frac{e^{- 6 i x}}{6 i} + 2 x) + C

= (\frac{e^{6 i x} - e^{- 6 i x}}{24 i}) + \frac{x}{2} + C

Thus the integral is obtained to be

(\frac{e^{6 i x} - e^{- 6 i x}}{24 i}) + \frac{x}{2} + C

.

Evaluate the integral. \int \cos^{2}3x dx

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