91Ó°ÊÓ

To find the antiderivative of \(\sec^4\theta\), we will apply integration by parts. Let's use the identity \(\sec^2\theta=1+\tan^2\theta\) and rewrite \(\sec^4\theta\) as \(\sec^2\theta\sec^2\theta\). Now, let \(u=\sec^2\theta\), \(dv=\sec^2\theta d\theta\), then \(du=2\sec^2\theta\tan\theta d\theta\) and \(v=\tan\theta\). Applying integration by parts: \(\int \sec^4\theta d\theta = \int u dv = uv - \int v du\) \(\int \sec^4\theta d\theta = \sec^2\theta\tan\theta - \int \tan\theta(2\sec^2\theta\tan\theta) d\theta\) Now, integrate the remaining integral: \(\int \sec^4\theta d\theta = \sec^2\theta\tan\theta - 2\int \tan^2\theta\sec^2\theta d\theta\) Use \(\sec^2\theta=1+\tan^2\theta\) and rewrite the integral: \(\int \sec^4\theta d\theta = \sec^2\theta\tan\theta - 2\int (1+\tan^2\theta-1)\sec^2\theta d\theta\) \(\int \sec^4\theta d\theta = \sec^2\theta\tan\theta - 2\int \tan^2\theta\sec^2\theta d\theta\) Now, let's evaluate the inner integral with the given limits.

Evaluate the inner integral with the given limits, from \(1\) to \(x\): \(\int_{1}^{x} \sec ^{4}\theta d\theta = (\sec^2x\tan x - 2\int_{1}^{x} \tan^2\theta\sec^2\theta d\theta) - (\sec^21\tan 1 - 2\int_{1}^{1} \tan^2\theta\sec^2\theta d\theta)\) Since the integral from \(1\) to \(1\) is \(0\), the expression becomes: \(\int_{1}^{x} \sec ^{4}\theta d\theta = \sec^2x\tan x - \sec^21\tan 1 - 2\int_{1}^{x} \tan^2\theta\sec^2\theta d\theta\) Next, we need to differentiate this expression with respect to \(x\).

Now, differentiate the expression with respect to \(x\): \(\frac{d}{dx}\left\\{\int_{1}^{x} \sec ^{4}\theta d\theta\right\\} = \frac{d}{dx}(\sec^2x\tan x - \sec^21\tan 1 - 2\int_{1}^{x} \tan^2\theta\sec^2\theta d\theta)\) Using the properties of derivatives, we have: \(\frac{d}{dx}(\sec^2x\tan x - \sec^21\tan 1 - 2\int_{1}^{x} \tan^2\theta\sec^2\theta d\theta) = (\sec^2x\sec^2x + \sec^2x\tan x) - 0 - (2\tan^2x\sec^2x)\) Simplify the expression: \(\frac{d}{dx}\left\\{\int_{1}^{x} \sec ^{4}\theta d\theta\right\\} = \sec^4x\) Finally, we need to evaluate the outer definite integral.

Now, evaluate the outer definite integral with the given limits, from \(0\) to \(\frac{\pi}{4}\): \(\int_{0}^{\frac{\pi}{4}} \frac{d}{d x}\left\\{\int_{1}^{x} \sec ^{4}\theta d \theta\right\\} d x = \int_{0}^{\frac{\pi}{4}}\sec^4x dx\) Calculate the integral: \(\int_{0}^{\frac{\pi}{4}}\sec^4x dx = \left( \sec^2\frac{\pi}{4}\tan \frac{\pi}{4} - \sec^21\tan 1 - 2\int_{1}^{\frac{\pi}{4}} \tan^2\theta\sec^2\theta d\theta \right) - \left( \sec^20\tan 0 - \sec^21\tan 1 - 2\int_{1}^{0} \tan^2\theta\sec^2\theta d\theta \right)\) Since \(\tan 0=0\), the expression becomes: \(\int_{0}^{\frac{\pi}{4}}\sec^4x dx = \sec^2\frac{\pi}{4}\tan \frac{\pi}{4} - 2\int_{1}^{\frac{\pi}{4}} \tan^2\theta\sec^2\theta d\theta\) After evaluating the integral and simplifying the expression, we have the final answer. \(\int_{0}^{\frac{\pi}{4}}\sec^4x dx = \boxed{\sec^2\frac{\pi}{4}\tan \frac{\pi}{4} - 2\int_{1}^{\frac{\pi}{4}} \tan^2\theta\sec^2\theta d\theta}\)