Problem 82 Calculate $$\int \sin x \cos x d... [FREE SOLUTION]

Chapter 5: Problem 82

Calculate $$\int \sin x \cos x d x$$ (a) Setting $u=\sin x$ (b) Setting $u=\cos x$ (c) Reconcile your answers to parts (a) and (b).

Short Answer

Expert verified

The integral of $\sin x \cos x$ with respect to $x$ can be solved using substitution. When setting $u = \sin x$, the integral is $\frac{1}{2}\sin^2 x + C_1$. When setting $u = \cos x$, the integral is $\frac{1}{2}\cos^2 x + C_2$. Although these integrals appear different, they differ by a constant factor and are therefore equivalent.

Step by step solution

Substitution Method: Setting u = sin(x)

(a) To substitute sin(x) with u, let's define: $u = \sin x$ Now, we need to find the differential of u with respect to x which is: $\frac{d u}{d x} = \cos x$ From this, we can write the differential dx in terms of du: $d x = \frac{d u}{\cos x}$ Now, let's replace sin(x) and dx in the integral with the corresponding expressions in terms of u: $\int \sin x \cos x d x = \int u \cos x \frac{d u}{\cos x}$ The cos(x) terms will cancel each other out, leaving us with: $\int u d u$ Now we can integrate u with respect to u: $\int u d u = \frac{1}{2}u^2 + C$ Now, let's replace u with sin(x) to get our final answer in terms of x: $\frac{1}{2}u^2 + C = \frac{1}{2}\sin^2 x + C_1$

Substitution Method: Setting u = cos(x)

(b) To substitute cos(x) with u, let's define: $u = \cos x$ Now, we need to find the differential of u with respect to x which is: $\frac{d u}{d x} = -\sin x$ From this, we can write the differential dx in terms of du: $d x = \frac{d u}{-\sin x}$ Now, let's replace cos(x) and dx in the integral with the corresponding expressions in terms of u: $\int \sin x \cos x d x = \int (-\sin x) u \frac{d u}{-\sin x}$ The -sin(x) terms will cancel each other out, leaving us with: $\int u d u$ Now we can integrate u with respect to u: $\int u d u = \frac{1}{2}u^2 + C$ Now, let's replace u with cos(x) to get our final answer in terms of x: $\frac{1}{2}u^2 + C = \frac{1}{2}\cos^2 x + C_2$

Reconciling Answers from Parts (a) and (b)

(c) Notice that the integrals in parts (a) and (b) are not equal. However, they differ by a constant factor. Let's find the relationship between them: $C_1 = \frac{1}{2}\sin^2 x + C$ $C_2 = \frac{1}{2}\cos^2 x + C$ Sum the two equations to eliminate C: $C_1 + C_2 = \frac{1}{2}(\sin^2 x + \cos^2 x) + 2C$ Given that $\sin^2 x + \cos^2 x = 1$, we have: $C_1 + C_2 = \frac{1}{2}(1) + 2C$ Hence, the two integrals are related by the following equation: $\frac{1}{2}(\sin^2 x + \cos^2 x) + C_1 = \frac{1}{2}(\sin^2 x + \cos^2 x) + C_2$ The two solutions differ by some constant factor, and therefore, they are equivalent.

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Key Concepts

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

Substitution Method

The substitution method is a very handy technique when integrating functions that seem too complicated at first glance. The basic idea is to replace a part of the integral with a new variable, often simplifying the integration process. Given an integral like $ \int \sin x \cos x \ dx $, it can be beneficial to use substitution to make the integral more manageable.

**Choosing the Substitution**: This step is crucial. The best substitution makes the complex part of the integral much simpler. For example, in our problem, setting $ u = \sin x $ or $ u = \cos x $ are both viable substitutions because they relate directly to the original function components.

**Differential Transformation**: Once you choose $ u $, find $ du/dx $ to express $ dx $ in terms of $ du $. This involves basic differentiation of the chosen $ u $.

**Simplifying the Integral**: Substitute $ u $ and its differential into the integral. Often, factors cancel, simplifying the integral greatly.

**Integrate and Back Substitute**: After integration, replace $ u $ with the original expression in terms of $ x $ to get the final answer in the initial variable.

This method is incredibly useful with functions that include composite expressions, as it breaks down the integral into more manageable pieces.

Trigonometric Integration

Trigonometric integration focuses on integrating functions involving trigonometric functions like sine and cosine. In our example $ \int \sin x \cos x \ dx $, several trigonometric identities can simplify the integration process, especially when dealing directly with these types of functions.

**Product of Sine and Cosine**: Integrals that involve products of sine and cosine are common in trigonometry. The formula $ \sin 2x = 2\sin x \cos x $ is particularly useful. Using identities can often reduce the work needed or present alternative solutions.

**Even and Odd Powers**: When dealing with even or odd powers of sine and cosine, particular strategies such as using half-angle formulas can be helpful. For instance, expressing a sine squared or cosine squared function using half-angle identities can reveal simpler forms.

**Flexibility in Methods**: Often, more than one approach can be taken - either using identities or substitution as seen in this example. This flexibility is key when tackling complex integrals.

Mastering trigonometric integration involves knowing identities and the typical forms these integrals take, often reviewing them both algebraically and visually to understand their transformations.

Definite and Indefinite Integrals

Integrals come in two flavors: definite and indefinite. Understanding the distinction and when to use them is vital in calculus.

**Indefinite Integrals**: These represent a family of functions and do not have bounds. They include an arbitrary constant $ C $, representing the vertical shift of the anti-derivative function. Integrating $ \sin x \cos x $ using substitution leads to an indefinite integral, characterized by a solution plus the constant $ C $.

**Definite Integrals**: These provide a number instead of a function, bounding the integral over an interval $[a, b]$. When you compute a definite integral, you aren't just finding a general solution, but instead, you're calculating the area under the curve within the specified limits.

**Physical Interpretation**: While indefinite integrals can be seen as finding anti-derivatives, definite integrals relate to accumulated quantities, such as areas under curves or total displacement given a velocity function.

These concepts are foundational in numerous mathematical applications, from basic physics to engineering, underscoring the importance of mastery in integration techniques.

91影视

Calculate $$\int \sin x \cos x d x$$ (a) Setting \(u=\sin x\) (b) Setting \(u=\cos x\) (c) Reconcile your answers to parts (a) and (b).

Short Answer

Step by step solution

Substitution Method: Setting u = sin(x)

Substitution Method: Setting u = cos(x)

Reconciling Answers from Parts (a) and (b)

Key Concepts

Substitution Method

Trigonometric Integration

Definite and Indefinite Integrals

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