Problem 20 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

zeigen sie die Richtigkeit aieser pormeln. BEISPIEL. - Gesucht ist $$ \int \frac{1}{1+\sin x} d x,-\frac{\pi}{2}

Step-by-Step Solution

Answer

The integral is \( -\frac{2}{1+\tan \frac{x}{2}} + C \).

Step 1: Understand the Problem

We need to find the indefinite integral \( \int \frac{1}{1+\sin x} \, dx \) in the given interval \( -\frac{\pi}{2}

Step 2: Apply the Substitution

Substitute \( u = \tan \frac{x}{2} \). Use the identity \( \sin x = \frac{2u}{1+u^2} \). Thus, \( 1 + \sin x = 1 + \frac{2u}{1+u^2} = \frac{1+u^2+2u}{1+u^2} \).

Step 3: Rewrite the Integral

Express the integral in terms of \( u \): \( \int \frac{1}{1+\sin x} \, dx = \int \frac{1+u^2}{1+u^2+2u} \, d \left(\frac{2}{1+u^2}\right) \, du \). This simplifies to \( \int \frac{2}{1+u^2+2u} \, du \).

Step 4: Simplify the Denominator

Factor the denominator: \( 1+u^2+2u = (u+1)^2 \). This transforms the integral into \( \int \frac{2}{(u+1)^2} \, du \).

Step 5: Integrate

Integrate \( \int \frac{2}{(u+1)^2} \, du \). The solution is \( -\frac{2}{u+1} + C \), where \( C \) is the constant of integration.

Step 6: Substitute Back

Replace \( u \) with \( \tan \frac{x}{2} \) to revert to the variable \( x \): \( -\frac{2}{1+\tan \frac{x}{2}} + C \).

Key Concepts

Substitution

Substitution is a powerful technique in calculus, particularly useful for solving integrals. The goal is to transform a complicated integral into a simpler one by changing the variable. In this example, we deal with the integral of \( \int \frac{1}{1+\sin x} \, dx \). This integral can be challenging directly, so we use the substitution \( u = \tan \frac{x}{2} \).

When we make this substitution, the function \( \sin x \) gets rewritten in terms of \( u \), using the identity \( \sin x = \frac{2u}{1+u^2} \). As a result, \( 1 + \sin x \) becomes \( \frac{1+u^2+2u}{1+u^2} \), effectively altering the integral into a more manageable form. This substitution allows us to factor the denominator and simplify the expression, easing the path to finding the integral.

Substitution is advantageous when it reduces complex trigonometric expressions into polynomial or more straightforward forms. In practice, always consider what substitution could reveal simpler expressions, transforming the original problem into something more familiar and easier to handle.

Trigonometrische Funktionen

Trigonometrische Funktionen, or trigonometric functions, are central to many problems in calculus. They include functions such as \( \sin x \), \( \cos x \), and \( \tan x \). In integral calculus, these functions often appear in challenging expressions that are difficult to integrate directly.

In our example, the use of the sine function, \( \sin x \), complicates the integral. However, by using the substitution \( u = \tan \frac{x}{2} \), we can reframe \( \sin x \) into a simpler rational form \( \frac{2u}{1+u^2} \).

• This substitution connects the trigonometric function to a more manageable polynomial form.
• The simplification transforms the function into a rational equation, making it easier to integrate.

This example illustrates how knowledge of trigonometric identities and their properties can facilitate solving integrals. Understanding these transformations is crucial for handling trigonometric integrals efficiently.

Unbestimmte Integrale

Unbestimmte Integrale, or indefinite integrals, represent a family of functions whose derivative is the given function. They are expressed without upper and lower limits and include a constant of integration, \( C \).

In the given exercise, we find the indefinite integral \( \int \frac{1}{1+\sin x} \, dx \) by transforming it through substitution. This results in the simpler integral \( \int \frac{2}{(u+1)^2} \, du \), which we can now solve straightforwardly.

The solution is \(-\frac{2}{u+1} + C \). Importantly, once the integration is completed, we substitute back to the original variable, \( x \), finishing with \(-\frac{2}{1+\tan \frac{x}{2}} + C \).

• Indefinite integrals provide a general solution including all possible solutions with the constant \( C \).
• They help understand the broader picture where specific definite integrals are particular instances.

This characteristic highlights the breadth of possibilities when working with integrals, allowing for flexibly addressing various boundary or initial conditions.

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Problem 17

Next

Problem 21

Other exercises in this chapter

Problem 13

Berechnen sie a) \(\int_{0}^{\frac{\pi}{2}} \cos ^{2} x d x\) b) \(\int_{0}^{\frac{\pi}{2}} x \sin x \cos x d x\) Manchmal vereinfacht die partielle Integration

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Problem 17

Bestimen sie mit der Substitutionsregel a) \(f(a x+b)^{n} d x\) b) \(\int \theta \cos \theta^{2} d \theta\) c) \(\int\left(\cos x-\cos ^{3} x\right) d x\)

View solution

Problem 21

Bestimimen sie $$ \int \frac{1}{\sin x} d x $$

View solution

Problem 22

AUPGABE 22. - Bestimen Sie a) \(\int_{1}^{2} e^{3 x} d x\) b) \(\int_{0}^{\frac{\pi}{2}} \sin \theta \cos ^{2} \theta d \theta .\) c) Bestimmen Sie den Flacheni

View solution

Key Concepts

Substitution

Substitution is a powerful technique in calculus, particularly useful for solving integrals. The goal is to transform a complicated integral into a simpler one by changing the variable. In this example, we deal with the integral of \( \int \frac{1}{1+\sin x} \, dx \). This integral can be challenging directly, so we use the substitution \( u = \tan \frac{x}{2} \).

When we make this substitution, the function \( \sin x \) gets rewritten in terms of \( u \), using the identity \( \sin x = \frac{2u}{1+u^2} \). As a result, \( 1 + \sin x \) becomes \( \frac{1+u^2+2u}{1+u^2} \), effectively altering the integral into a more manageable form. This substitution allows us to factor the denominator and simplify the expression, easing the path to finding the integral.

Substitution is advantageous when it reduces complex trigonometric expressions into polynomial or more straightforward forms. In practice, always consider what substitution could reveal simpler expressions, transforming the original problem into something more familiar and easier to handle.

Trigonometrische Funktionen

Trigonometrische Funktionen, or trigonometric functions, are central to many problems in calculus. They include functions such as \( \sin x \), \( \cos x \), and \( \tan x \). In integral calculus, these functions often appear in challenging expressions that are difficult to integrate directly.

In our example, the use of the sine function, \( \sin x \), complicates the integral. However, by using the substitution \( u = \tan \frac{x}{2} \), we can reframe \( \sin x \) into a simpler rational form \( \frac{2u}{1+u^2} \).

• This substitution connects the trigonometric function to a more manageable polynomial form.
• The simplification transforms the function into a rational equation, making it easier to integrate.

This example illustrates how knowledge of trigonometric identities and their properties can facilitate solving integrals. Understanding these transformations is crucial for handling trigonometric integrals efficiently.

Unbestimmte Integrale

Unbestimmte Integrale, or indefinite integrals, represent a family of functions whose derivative is the given function. They are expressed without upper and lower limits and include a constant of integration, \( C \).

In the given exercise, we find the indefinite integral \( \int \frac{1}{1+\sin x} \, dx \) by transforming it through substitution. This results in the simpler integral \( \int \frac{2}{(u+1)^2} \, du \), which we can now solve straightforwardly.

The solution is \(-\frac{2}{u+1} + C \). Importantly, once the integration is completed, we substitute back to the original variable, \( x \), finishing with \(-\frac{2}{1+\tan \frac{x}{2}} + C \).

• Indefinite integrals provide a general solution including all possible solutions with the constant \( C \).
• They help understand the broader picture where specific definite integrals are particular instances.

This characteristic highlights the breadth of possibilities when working with integrals, allowing for flexibly addressing various boundary or initial conditions.