Problem 20 Show that \(t=\tan \frac{1}{2} x... [FREE SOLUTION]

Chapter 8: Problem 20

Show that $t=\tan \frac{1}{2} x$ implies $$ \begin{aligned} &\sin x=\frac{2 t}{1+t^{2}} \\ &\cos x=\frac{1-t^{2}}{1+t^{2}} \end{aligned} $$ and $$ \mathrm{d} x=\frac{2}{1+t^{2}} \mathrm{~d} t $$ Hence integrate (a) $\operatorname{cosec} x$ (b) $\sec x$ (c) $\frac{1}{3+4 \sin x}$ (d) $\frac{1}{5 \sin x+12 \cos x}$

Short Answer

Expert verified

The expressions for sin, cos, and dx in terms of t are verified, and the integrals are solved using substitution and partial fractions.

Step by step solution

Expressing sin x and cos x in terms of t

Since $ t = \tan \frac{1}{2}x $, we can use the identities: $ \sin x = 2\tan\frac{1}{2}x / (1 + \tan^2 \frac{1}{2}x)$ and $ \cos x = (1 - \tan^2 \frac{1}{2}x) / (1 + \tan^2 \frac{1}{2}x)$. Substituting $ t $ in place of $ \tan\frac{1}{2}x $, we get $ \sin x = \frac{2t}{1+t^2} $ and $ \cos x = \frac{1-t^2}{1+t^2} $.

Deriving dx in terms of dt

To find $ \mathrm{d}x $, start from $ t = \tan \frac{1}{2}x $. Taking the derivative with respect to $ t $, we use the inverse tangent differentiation: $ \frac{1}{2} \frac{d}{dx}(x) = \frac{1}{1+t^2} $ or $ \mathrm{d}x = \frac{2}{1+t^2} \mathrm{d}t $.

Integrating $\operatorname{cosec} x$

$ \operatorname{cosec} x = \frac{1}{\sin x} $, using $ \sin x = \frac{2t}{1+t^2} $, we find $ \operatorname{cosec} x = \frac{1+t^2}{2t} $. Thus, the integral becomes $ \int \operatorname{cosec} x \mathrm{d}x = \int \frac{1+t^2}{2t} \cdot \frac{2}{1+t^2} \mathrm{d}t = \int \frac{1}{t} \mathrm{d}t $. The result is $ \ln |t| + C $, or $ \ln |\tan \frac{1}{2}x| + C $.

Integrating $\sec x$

$ \sec x = \frac{1}{\cos x} $, using $ \cos x = \frac{1-t^2}{1+t^2} $, the integral becomes $ \int \sec x \mathrm{d}x = \int \frac{1+t^2}{1-t^2} \cdot \frac{2}{1+t^2} \mathrm{d}t = \int \frac{2}{1-t^2} \mathrm{d}t $. This can be integrated using partial fraction decomposition, resulting in $ \ln \left| \frac{1+t}{1-t} \right| + C $.

Integrating $\frac{1}{3+4 \sin x}$

Substituting $ \sin x = \frac{2t}{1+t^2} $, the integral becomes $ \int \frac{1+t^2}{3(1+t^2) + 8t}\mathrm{d}t $. Solving by partial fractions yields a fraction decomposition of $ \frac{1}{8} \left( \ln |3+8t+t^2| \right) + C $.

Integrating $\frac{1}{5 \sin x + 12 \cos x}$

Substituting $ \sin x = \frac{2t}{1+t^2} $ and $ \cos x = \frac{1-t^2}{1+t^2} $, the integral becomes $ \int \frac{1+t^2}{5(2t) + 12(1-t^2)} \mathrm{d}t $. This simplifies to $ \int \frac{1+t^2}{12-t^2} \mathrm{d}t $ and is integrated through partial fractions to get $ \frac{1}{2\sqrt{3}} \ln \left| \frac{t + \sqrt{3}}{t - \sqrt{3}} \right| + C $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities

In trigonometry, identities are equations that hold true for all values of the variable involved. One of the key trigonometric substitutions used in integration problems is replacing $t = \tan \frac{1}{2}x$. This substitution transforms complex trigonometric functions into more manageable form.

For $\sin x$, we can use the identity: $\sin x = \frac{2t}{1+t^2}$. This represents the sine function using our variable $t$.
Similarly, $\cos x = \frac{1-t^2}{1+t^2}$ helps in expressing cosine in terms of $t$.

These simplified expressions are extremely helpful when you are integrating trigonometric functions using substitution as it reduces the complexity of the integrand.
Knowing these identities allows us to transform integration problems to a polynomic form which is easier to integrate. This is a crucial part of understanding integration techniques.

Integration Techniques

Integration is a fundamental concept in calculus, and various techniques can be used to find integrals. When dealing with trigonometric functions, substitution is a common strategy.
By substituting $t = \tan \frac{1}{2}x$, you can express $(\sin x, \cos x, \text{and} \mathrm{d} x)$ in terms of $t$. This transforms integrals involving trigonometric functions into simpler rational expressions.

Consider $\operatorname{cosec} x$, transformed into $\int \frac{1}{t} \mathrm{d}t$, which is straightforward to integrate into $\ln |t| + C$.
Similarly, $\sec x$ becomes $\int \frac{2}{1-t^2} \mathrm{d}t$, allowing for partial fraction decomposition.

By changing variables, you can utilize simpler algebraic techniques to perform integration, streamlining the process and enhancing understanding.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to simplify the integration of rational expressions. This technique becomes highly effective when integrating functions that result from trigonometric substitutions.
Consider the integral $\int \frac{2}{1-t^2} \mathrm{d}t$. To solve this, you can rewrite it using the partial fraction decomposition approach:

The expression can be represented as $\int \left( \frac{A}{1+t} + \frac{B}{1-t} \right) \mathrm{d}t$, where $A$ and $B$ are constants.
Solving for $A$ and $B$, you recompose the integral into separate terms, each of which can be integrated easily.

This technique streamlines the integration process, converting complex expressions into sums of easier standard forms.

Calculus Problems

Calculus problems often involve integrating complex expressions. Understanding trigonometric identities, substitution, and decomposition techniques is essential.

For expressions like $\frac{1}{5 \sin x + 12 \cos x}$, substituting trigonometric identities simplifies the problem.
These expressions often result in more friendly algebraic forms that can further be decomposed using partial fractions.

Breaking down these problems step by step ensures you see how trigonometric principles are applied and simplifies solving calculus problems effectively. This level of understanding allows tackling a wide range of calculus problems more efficiently.

91影视

Short Answer

Step by step solution

Expressing sin x and cos x in terms of t

Deriving dx in terms of dt

Integrating \(\operatorname{cosec} x\)

Integrating \(\sec x\)

Integrating \(\frac{1}{3+4 \sin x}\)

Integrating \(\frac{1}{5 \sin x + 12 \cos x}\)

Key Concepts

Trigonometric Identities

Integration Techniques

Partial Fraction Decomposition

Calculus Problems

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