Find the trigonometric integral ∫sec4(x2)dx

Question

grbavit · Accepted Answer

Step 1
To evaluate the trigonometric integral,

\int \sec^{4} (\frac{x}{2}) d x

Solution:
Put

\frac{x}{2} = t

in the given integral we get,
dx=2dt
And the integral becomes,

2 \int \sec^{4} (t) d t

Solving the above integral as,
Step 2

2 \int \sec^{4} (t) d t = 2 \int \sec^{2} (t) \sec^{2} (t) d t

= 2 \int \sec^{2} (t) [1 + \tan^{2} t] d t

Again use substitution method as,
Put

\tan t = u

\sec^{2} t d t = d u

The integral becomes,

2 \int \sec^{2} (t) [1 + \tan^{2} t] d t = 2 \int (1 + u^{2}) d u

= 2 (u + \frac{u^{3}}{3}) + c

= 2 u + 2 \frac{u^{3}}{3} + c

Step 3
Since

u = \tan t and t = \frac{x}{2}

, the integral gives,

\int \sec^{4} (\frac{x}{2}) d x = 2 (\tan \frac{x}{2}) + \frac{2}{3} {(\tan \frac{x}{2})}^{3} + c

Hence, the value of the integral is

2 (\tan \frac{x}{2}) + \frac{2}{3} {(\tan \frac{x}{2})}^{3} + c

.

Find the trigonometric integral \int \sec^{4}(\frac{x}{2})dx