Problem 57 - Thomas Calculus

Question

Evaluate each integral in Exercises \(57-62\) by multiplying by a form of 1 and using a substitution (if necessary) to reduce it to standard form. $$ \int \frac{1}{1+\sin x} d x $$

Step-by-Step Solution

Answer

The integral is \( \tan x - \sec x + C \).

Step 1: Rationalizing the Denominator

Multiply the integrand by \( \frac{1-\sin x}{1-\sin x} \), which is a form of 1, to eliminate \( \sin x \) from the denominator. This gives:\[ \int \frac{1-\sin x}{(1+\sin x)(1-\sin x)} \, dx = \int \frac{1-\sin x}{1-\sin^2 x} \, dx \]

Step 2: Simplify Using Trigonometric Identity

Using the identity \( 1 - \sin^2 x = \cos^2 x \), the integral becomes: \[ \int \frac{1-\sin x}{\cos^2 x} \, dx \] Break this into two separate integrals:\[ \int \frac{1}{\cos^2 x} \, dx - \int \frac{\sin x}{\cos^2 x} \, dx \]

Step 3: Integrate Each Term Separately

For the first integral, recall \( \frac{1}{\cos^2 x} = \sec^2 x \): \[ \int \sec^2 x \, dx = \tan x + C_1 \]For the second integral, use substitution. Let \( u = \cos x \), then \( du = -\sin x \, dx \) or \( -du = \sin x \, dx \):\[ -\int \frac{-du}{u^2} = \int u^{-2} \, du = -u^{-1} + C_2 \]

Step 4: Substitute Back the Variable

Substitute \( u = \cos x \) back into the expression:\[ -u^{-1} = -\frac{1}{\cos x} = -\sec x \]Thus, the second integral becomes:\[ -\sec x + C_2 \]

Step 5: Combine the Results

Combine the results from Steps 3 and 4:\[ \int \frac{1}{1+\sin x} \, dx = \tan x - \sec x + C \] where \( C = C_1 + C_2 \).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is key to solving calculus problems involving trigonometric functions, such as the given integral. A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variable where both sides of the equality are defined. These identities help simplify complex expressions.For example, in our solution, we used the Pythagorean identity: \[ 1 - \sin^2 x = \cos^2 x \]This identity is useful because it allows us to replace the expression \(1 - \sin^2 x\) with \(\cos^2 x\), simplifying our integrand.Some other important trigonometric identities include:

• \(\sin^2 x + \cos^2 x = 1\)
• \(1 + \tan^2 x = \sec^2 x\)
• \(1 + \cot^2 x = \csc^2 x\)

By mastering these identities, you can efficiently tackle and simplify integrals, making the integration process smoother.

Integration by Substitution

Integration by substitution is a powerful technique often used in calculus to evaluate integrals. It's similar to the chain rule for differentiation and works well when a function inside an integral is the derivative of some other function. In such cases, substitution can simplify the integral into a form that is easier to work with.In our solution, substitution was used in Step 3 for the second integral \(\int \frac{\sin x}{\cos^2 x} \, dx\). Here, we made the substitution:

• Let \( u = \cos x \)
• Then, \( du = -\sin x \, dx \) or \(-du = \sin x \, dx\)

This transformed the integral into:\[ \int \frac{-du}{u^2} = \int u^{-2} \, du = -u^{-1} + C_2 \]After substitution, the integral became much simpler to solve, reducing it to a basic power rule application. Once solved, we substituted back the original variable, revealing the final result. Mastery of substitution is vital for solving various types of integrals, especially those involving functions and their derivatives.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate irrational elements, such as square roots, from the denominator of a fraction. In the context of calculus integration, this method can also be applied when dealing with trigonometric integrals to make them more manageable.In the provided solution, this technique was deftly applied by multiplying the original function by \(\frac{1-\sin x}{1-\sin x}\). This adjustment is not just a trick but a strategic use of the identity:\[ \text{Form of 1: } \frac{1-\sin x}{1-\sin x} \]This operation allows the transformation \[ \frac{1}{1+\sin x} \rightarrow \frac{1-\sin x}{1^2-\sin^2 x} = \frac{1-\sin x}{\cos^2 x} \], where the denominator simplifies significantly using the identity \(1 - \sin^2 x = \cos^2 x\).By applying rationalization effectively, the resulting expression can often be integrated more efficiently, streamlining the process significantly in problems where trigonometric identities and simplifications are involved.