Question Statement

Evaluate the definite integral:

${\int }_0^{\frac{\pi }{4}}{\sec }^{2}2\theta d\theta$

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Background and Explanation

This problem requires evaluating a definite integral using the substitution method (u-substitution) to handle the composite function ${\sec }^2(2\theta )$. Note that this integral is improper because the integrand approaches infinity at the upper limit, which requires evaluating the behavior of the antiderivative at the boundary.

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Solution

Let

$I={\int }_0^{\frac{\pi }{4}}{\sec }^{2}2\theta d\theta$

Step 1: Substitution

Put $2\theta =t$. Then:

• $\theta =\frac{t}{2}$
• $d\theta =\frac{dt}{2}$

Step 2: Change the limits of integration

• When $\theta =0$: $t=2(0)=0$
• When $\theta =\frac{\pi }{4}$: $t=2\cdot \frac{\pi }{4}=\frac{\pi }{2}$

Step 3: Rewrite the integral

Substituting into equation (1):

$I={\int }_0^{\frac{\pi }{2}}{\sec }^{2}t\cdot \frac{dt}{2}$

$=\frac{1}{2}{\int }_0^{\frac{\pi }{2}}{\sec }^{2}tdt$

Step 4: Integrate

Using the standard integral $\int {\sec }^{2}tdt=\tan t$:

$I=\frac{1}{2}{\left[\tan t\right]}_0^{\frac{\pi }{2}}$

Step 5: Evaluate at the limits

$I=\frac{1}{2}\left(\tan \frac{\pi }{2}-\tan 0\right)$

Since $\tan \frac{\pi }{2}$ approaches infinity (is undefined) and $\tan 0=0$:

$I=\frac{1}{2}(\infty -0)$

$=\infty$

Conclusion: The integral is divergent (it does not converge to a finite value).

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Key Formulas or Methods Used

• Substitution rule for definite integrals: If $t=g(\theta )$, then ${\int }_a^{b}f(g(\theta ))g^{\prime }(\theta )d\theta ={\int }_{g(a)}^{g(b)}f(t)dt$
• Standard integral: $\int {\sec }^{2}xdx=\tan x+C$
• Improper integral evaluation: Recognizing when antiderivatives approach infinity at boundary points

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Summary of Steps

1. Set up the integral: $I={\int }_0^{\frac{\pi }{4}}{\sec }^{2}2\theta d\theta$
2. Apply substitution: Let $t=2\theta$, so $d\theta =\frac{dt}{2}$
3. Change integration limits: $0\to 0$ and $\frac{\pi }{4}\to \frac{\pi }{2}$
4. Rewrite: $I=\frac{1}{2}{\int }_0^{\frac{\pi }{2}}{\sec }^{2}tdt$
5. Integrate: $\frac{1}{2}[\tan t{]}_0^{\frac{\pi }{2}}$
6. Evaluate limits: $\frac{1}{2}(\tan (\frac{\pi }{2})-\tan (0))=\frac{1}{2}(\infty -0)$
7. Conclude the integral diverges to infinity