Question Statement

Evaluate the integral:

$\int \left(1+{\tan }^2\theta \right)d\theta$

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Background and Explanation

This problem requires familiarity with fundamental trigonometric identities relating tangent and secant functions, as well as basic integration rules for trigonometric expressions.

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Solution

Method 1: Term-by-Term Integration

We begin by splitting the integral into two separate terms using the linearity property:

$I=\int \left(1+{\tan }^2\theta \right)d\theta =\int 1d\theta +\int {\tan }^2\theta d\theta$

The first integral evaluates to $\theta$. For the second integral, we apply the trigonometric identity ${\tan }^2\theta ={\sec }^2\theta -1$:

$I=\theta +\int \left({\sec }^2\theta -1\right)d\theta$

Distributing the integral across the subtraction:

$I=\theta +\int {\sec }^2\theta d\theta -\int 1d\theta$

Evaluating each integral separately:

• $\int {\sec }^2\theta d\theta =\tan \theta$ (standard integral)
• $\int 1d\theta =\theta$

Substituting these results back:

$I=\theta +\tan \theta -\theta +c$

Simplifying by combining like terms ($\theta -\theta =0$):

$I=\tan \theta +c$

Method 2: Direct Identity Application

Alternatively, we can simplify the integrand immediately using the identity:

${\sec }^2\theta -{\tan }^2\theta =1\Rightarrow 1+{\tan }^2\theta ={\sec }^2\theta$

Therefore:

$I=\int \left(1+{\tan }^2\theta \right)d\theta =\int {\sec }^2\theta d\theta =\tan \theta +c$

Verification

To confirm our answer is correct, we differentiate the result:

$\frac{d}{d\theta }(\tan \theta +c)=\frac{d}{d\theta }(\tan \theta )+\frac{d}{d\theta }(c)={\sec }^2\theta +0=1+{\tan }^2\theta$

Since the derivative equals the original integrand, the solution $\tan \theta +c$ is verified.

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Key Formulas or Methods Used

• Pythagorean Identity: $1+{\tan }^2\theta ={\sec }^2\theta$ (or equivalently ${\sec }^2\theta -{\tan }^2\theta =1$)
• Basic Integral: $\int {\sec }^2\theta d\theta =\tan \theta +c$
• Basic Integral: $\int 1d\theta =\theta +c$
• Linearity of Integration: $\int [f(\theta )+g(\theta )]d\theta =\int f(\theta )d\theta +\int g(\theta )d\theta$

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Summary of Steps

1. Identify the integrand: Recognize $1+{\tan }^2\theta$ as a candidate for trigonometric simplification
2. Apply identity: Use $1+{\tan }^2\theta ={\sec }^2\theta$ to rewrite the integrand (or split the integral and substitute ${\tan }^2\theta ={\sec }^2\theta -1$)
3. Integrate: Evaluate $\int {\sec }^2\theta d\theta =\tan \theta$
4. Add constant: Include the constant of integration $+c$
5. Verify (optional): Differentiate the result to confirm it matches the original integrand