Problem 15

Question

Calculate. (If you run out of ideas, use the examples as models.) $$\int \sin 2 x \cos 3 x d x$$.

Step-by-Step Solution

Verified

Answer

The short answer is: $\int \sin(2x)\cos(3x) dx = -\frac{1}{10}\cos(5x)+\frac{1}{2}\cos(x)+C$

1Step 1: Apply Double-Angle Formula

We will apply the product-to-sum formulas for sine and cosine to simplify the function: $\sin(2x)\cos(3x) = \frac{1}{2}(\sin(5x) - \sin(x))$ Step 2: Integrate Individual Trigonometric Functions

2Step 2: Integrate Individual Trigonometric Functions

Now, we need to integrate these individual sine functions with respect to $x$: $\int (\frac{1}{2}\sin(5x) - \frac{1}{2}\sin(x)) dx$ Step 3: Apply Fundamental Integration Rules

3Step 3: Apply Fundamental Integration Rules

We will apply the Fundamental Integration Rule $\int \sin(ax)dx = -\frac{1}{a}\cos(ax)+C$: $-\frac{1}{10}\cos(5x)+\frac{1}{2}\cos(x)+C$ Step 4: Write Final Answer

4Step 4: Write Final Answer

The indefinite integral is: $-\frac{1}{10}\cos(5x)+\frac{1}{2}\cos(x)+C$

Key Concepts

Trigonometric IntegralsProduct-to-Sum FormulasFundamental Integration Rules

Trigonometric Integrals

Trigonometric integrals are when you integrate functions that involve trigonometric expressions like sine, cosine, tangent, and others. These can sometimes seem intimidating because they often involve several steps and require knowledge of various trigonometric identities. However, understanding the basics of these integrals makes the process much easier.

With trigonometric integrals, certain strategies are often applied to simplify the expression before integrating:

Using identities such as double-angle or half-angle formulas.
Breaking down products or other complex operations into simpler forms.

In our specific problem, the integral of $ \int \sin 2x \cos 3x \ dx $ involves a product of sine and cosine functions. Knowing the right identity to use helps turn this into an easier problem.

Product-to-Sum Formulas

The product-to-sum formulas are handy for converting products of sines and cosines into sums that are easier to work with, especially when integrating. These identities allow us to transform expressions into forms that are more straightforward to integrate.

For example, one useful formula is:

$ \sin(A)\cos(B) = \frac{1}{2} (\sin(A+B) + \sin(A-B)) $

Applying the product-to-sum formula simplifies the integration process. In our problem, $ \sin(2x)\cos(3x) $ is converted using:

$ \sin(2x)\cos(3x) = \frac{1}{2}(\sin(5x) - \sin(x)) $

This step is crucial as it breaks down the product into separate functions that we can then integrate individually.

Fundamental Integration Rules

The fundamental integration rules are the basic building blocks for calculus, allowing us to find antiderivatives of functions. When handling trigonometric integrals, applying these rules becomes crucial.

The rule for integrating sine functions, $ \int \sin(ax) \, dx = -\frac{1}{a}\cos(ax) + C $, is one of the fundamental rules used here.

Once the trigonometric product becomes a sum (thanks to the product-to-sum formula), each term like $ \sin(5x) $ and $ \sin(x) $ can be integrated separately using these rules.

In the example, after substitution, the integration applies:

$ \int \sin(5x) \, dx = -\frac{1}{5}\cos(5x) + C $
$ \int \sin(x) \, dx = -\cos(x) + C $

Combining these gives the final integral result: $-\frac{1}{10}\cos(5x)$ and $+\frac{1}{2}\cos(x) + C$.

Understanding these rules allows you to tackle trigonometric integrals confidently and is essential for learning more complex integration techniques.

Problem 15

Other exercises in this chapter

Problem 15

Determine the values of $n$ which guarante: a theoretical error less than $\epsilon$ if the integral is estimated by: $(a)$ the trapezoidal rule; (b) Simp

View solution

Problem 15

Calculate. $$\int \frac{x+3}{x^{2}-3 x+2} d x$$.

View solution

Problem 15

Calculate. $$\int \frac{x^{2}}{\left(x^{2}+8\right)^{3 / 2}} d x$$.

View solution

Problem 15

Calculate. $$\int \frac{d x}{\sqrt{1+e^{x}}}$$

View solution