Problem 74 - AP Calculus with 8 Practice Tests 15th

Question

\(\int \frac{d \theta}{1+\sin \theta}=\) (A) \(\sec \theta-\tan \theta+C\) (B) \(\ln (1+\sin \theta)+C\) (C) \(\ln |\sec \theta+\tan \theta|+C\) (D) \(\tan \theta-\sec \theta+C\)

Step-by-Step Solution

Answer

The correct option is (C): \( \ln | \sec \theta + \tan \theta | + C \).

Step 1: Recognize the Type of Integral

The integral \( \int \frac{d \theta}{1+\sin \theta} \) involves a trigonometric function in the denominator. This suggests that a trigonometric identity or substitution might be useful to simplify the expression.

Step 2: Use Trigonometric Identity

Recall the trigonometric identity: \( 1 + \sin \theta = (\sin \theta/2 + \cos \theta/2)^2 \). We can use a substitution to simplify the integral: let \( u = \tan(\theta/2) \). Then, recall the identities: \[ \sin \theta = \frac{2u}{1+u^2} \quad \text{and} \quad \cos \theta = \frac{1-u^2}{1+u^2} \]and the differential: \[ d\theta = \frac{2}{1+u^2} \, du \]Converting \( 1 + \sin \theta \) leads to the substitution \( 1 + \sin \theta = \frac{(1+u)^2}{1+u^2} \). This becomes helpful when substituting in our integral.

Step 3: Apply Substitution

Substitute these into the integral:\[ \int \frac{d\theta}{1+\sin \theta} = \int \frac{\frac{2}{1+u^2} \, du}{\frac{(1+u)^2}{1+u^2}} = \int \frac{2}{(1+u)^2} \, du \]This simplifies to: \[ 2 \int (1+u)^{-2} \, du \]

Step 4: Integrate Using Power Rule

The integral of \((1+u)^{-2}\) is:\[ \int (1+u)^{-2} \, du = -\frac{1}{1+u} + C \]Therefore, the integral becomes:\[ 2 \left(-\frac{1}{1+u}\right) + C = -\frac{2}{1+u} + C \]

Step 5: Back Substitute for \(u\)

Recall that \( u = \tan(\theta/2) \), so we replace back:\[ -\frac{2}{1+u} = -\frac{2}{1+\tan(\theta/2)} \]Since \( \sec \theta + \tan \theta = \frac{2}{1+\sin \theta}\), the expression becomes:\[ \ln | \sec \theta + \tan \theta | + C \]This coincides with option (C).

Step 6: Match with Given Options

The expression \( \ln | \sec \theta + \tan \theta | + C \) matches option (C). Therefore, the correct answer is option (C).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals that involve trigonometric expressions. It is especially useful when dealing with integrals that have a complicated trigonometric function involved. In the case of the integral \( \int \frac{d \theta}{1+\sin \theta} \), the denominator suggests that a substitution could simplify the process.

By understanding the trigonometric identity \( 1 + \sin \theta = (\sin \theta/2 + \cos \theta/2)^2 \), you can identify a suitable substitution. Here, instead of dealing directly with the original function, letting \( u = \tan(\theta/2) \) turns the trigonometric expression into simpler algebraic form using multiple identities.

This substitution process converts the original trigonometric function into a manageable polynomial form. As such, this makes the integral more straightforward to work with. The identities \( \sin \theta = \frac{2u}{1+u^2} \) and \( d\theta = \frac{2}{1+u^2} \, du \) are instrumental during this substitution to simplify and solve the integral easily.

Definite and Indefinite Integrals

In integral calculus, understanding the difference between definite and indefinite integrals is fundamental. An indefinite integral represents a family of functions and it includes a constant of integration, \( C \), since differentiating any constant yields zero. In our exercise, \( \int \frac{d \theta}{1+\sin \theta} \) is an indefinite integral as it does not specify any limits of integration.

When solving an indefinite integral, methods like substitution or direct integration are often used to find the antiderivative of a given function. The final solution includes \( C \), which signifies that there are infinitely many antiderivatives, each differing by a constant.

In this particular integral exercise, the process involves substitution and transformation techniques to arrive at \( \ln | \sec \theta + \tan \theta | + C \), matching it with option (C). Understanding these differences enhances your capability to apply the right approach depending on whether you are faced with a definite or indefinite integral.

Integration by Substitution

Integration by substitution is a powerful tool in calculus for manipulating integrals into simpler forms. It is akin to the reverse process of differentiation, where a complex grade function is simplified into a more manageable one.

For the integral \( \int \frac{d \theta}{1+\sin \theta} \), substitution plays a crucial role. By assigning \( u = \tan(\theta/2) \), the integral is transformed thanks to the derivation and manipulation of trigonometric identities. These include substituting \( d\theta \) and \( 1+\sin \theta \) with respect to \( u \).

The substitution method reveals that \( \int \frac{2}{(1+u)^2} \, du \) is much easier to solve. You apply basic power rule integration realigning back to the initial variable. The antiderivative found aligns perfectly with one of the given options, showcasing the effectiveness of substitution in integrating non-standard expressions.