Problem 79

Question

Let $m$ and $n$ be integers such that $m \neq-n$ Prove the $$ \begin{aligned} \int \sin ^{n}(x) \cos ^{m}(x) d x=&-\frac{\sin ^{n-1}(x) \cos ^{m+1}(x)}{m+n} \\\ &+\frac{n-1}{m+n} \int \sin ^{n-2}(x) \cos ^{m}(x) d x \end{aligned} $$.

Step-by-Step Solution

Verified

Answer

Apply integration by parts and adjust for $ m+n $ using trigonometric manipulations.

1Step 1: Break down the integral

Consider the integral $ I = \int \sin^{n}(x) \cos^{m}(x)\, dx $. We'll use integration by parts to simplify this integral.

2Step 2: Choose parts for integration by parts

Integration by parts formula is $ \int u\, dv = uv - \int v\, du $. Let $ u = \sin^{n-1}(x) $ and $ dv = \sin(x)\cos^{m}(x)\, dx $.

3Step 3: Differentiate and integrate

Compute $ du = (n-1)\sin^{n-2}(x)\cos(x)\, dx $ and $ v = -\frac{\cos^{m+1}(x)}{m+1} $ by substituting $ dv $ as $ \cos(x)d(\sin(x)) $ and integrating.

4Step 4: Apply the integration by parts formula

Substitute to get $ I = uv - \int v\, du $. This leads to the expression \[-\frac{\sin^{n-1}(x)\cos^{m+1}(x)}{m+1} + \frac{n-1}{m+1} \int \sin^{n-2}(x)\cos^{m+1}(x)\, dx \].

5Step 5: Adjust for factor $ m+n $

Notice the need for $ m+n $ instead of $ m+1 $. Solve this by carefully analyzing and reorganizing the integral's structure through trigonometric identities or factor multiplication as needed. This manipulation aligns the integral with the given expression.

Key Concepts

Trigonometric IntegralsCalculus Problem SolvingIntegral Calculus Techniques

Trigonometric Integrals

One of the notable techniques in integral calculus is dealing with trigonometric integrals. These integrals often involve powers of trigonometric functions like sine and cosine,
and solving them requires familiarity with various identities and properties of trigonometric functions. When tackling an integral like $ \int \sin^{n}(x) \cos^{m}(x) \, dx $, understanding the behavior of sine and cosine as they oscillate is crucial.
Here are a few key points to remember:

**Symmetry**: Sine and cosine have even/odd properties useful in simplification.
**Periodic Nature**: Their periodic behavior sometimes allows replacing them in lengthy expressions using relations like $ \sin(x) = \cos(\frac{\pi}{2} - x) $.
**Pythagorean Identities**: Using $ \sin^2(x) + \cos^2(x) = 1 $ to express one trigonometric function in terms of the other can simplify the integral.

Finding a way to express one function in terms of the other or using identities can convert complex integrals into manageable forms. Remember that many integrals can be simplified by recognizing these fundamental traits of trigonometric functions.

Calculus Problem Solving

Calculus problem solving revolves around understanding concepts and choosing the right approach for each problem. Calculus, especially involving integrals, often requires a strategic selection of techniques. Let's highlight a typical scenario for solving the problem regarding integrals of trigonometric functions.
1. **Identify the Type of Integrals**: Recognize whether you are dealing with algebraic or trigonometric integrals, as each has different approaches.
2. **Select the Correct Method**:

**Integration by Parts**: Useful for products of functions, focusing on reducing powers and simplifying expressions.
**Trigonometric Substitutions**: Leveraged when expressions can be made simpler using sine, cosine, or tangent. Sometimes necessary when factors like $ \sin(x)\cos(x) $ need decomposing.

3. **Decompose Complex Expressions**: Use algebraic manipulations to simplify and reduce complexity, using identities and inverse functions when necessary.
Each problem requires careful scrutiny, and choosing an efficient method is the key. Always re-evaluate after integrating, especially when the initial assumptions of chosen parts could impact the overall result.

Integral Calculus Techniques

Integral calculus is full of techniques that simplify and enable us to solve diverse integrals. The technique of **Integration by Parts** is particularly crucial when dealing with products of functions.
The integration by parts formula is derived directly from the product rule for differentiation: $\int u \, dv = uv - \int v \, du.$This technique essentially transforms one integral into another in the hope that the new integral is simpler.When solving $ \int \sin^{n}(x) \cos^{m}(x) \, dx $ using integration by parts, a careful selection of functions for $ u $ and $ dv $ is necessary. Selecting wisely ensures simplification. Normal choices exploit the derivative properties:

Pick $ u $ as a function that simplifies when differentiated (e.g., powers of sine or cosine).
Choose $ dv $ that can be integrated into a function not increasing complexity (like basic trigonometric forms).

Remember to always inspect the constants or additional factors that arise, like $(m+n)$ in our original problem, as it requires readjusting the formula to fit the given constraints or identities.

Problem 78

Problem 79

Other exercises in this chapter

Problem 78

Show that the improper integral $\Gamma(s)=\int_{0}^{\infty} x^{s-1} e^{-x} d x$ is convergent for $s>0$. This function of $s$ is called the gamma functio

View solution

Problem 78

A region $\mathcal{R}$ is described. Calculate its area. $$ \begin{aligned} &\mathcal{R} \text { is the region that is bounded above by } y=\ln (1+x) \text {

View solution

Problem 79

Let $\alpha=\int_{0}^{1}\left(1+x^{4}\right)^{-1} d x .$ Express $\int_{0}^{1} \ln \left(1+x^{4}\right) d x$ in terms of $\alpha$. (It is known that \(\al

View solution

Problem 79

Make a (nontrigonometric) indirect substitution $x=\phi(u)$ to evaluate the given integral. $$ \int \frac{1}{1+\sqrt{x}} d x $$

View solution