https://r2.erweima.ai/mathbot/image/17255917598383s2pqo55.jpg

To solve the integral problem from the image, we need to evaluate the integral:

\[
\int (\sin 2x + \cos 5x) \, dx
\]

Let's break down the solution step-by-step:

### Step 1: Integrate \(\sin 2x\)
The integral of \(\sin 2x\) with respect to \(x\) is:

\[
\int \sin 2x \, dx = -\frac{1}{2} \cos 2x
\]

### Step 2: Integrate \(\cos 5x\)
The integral of \(\cos 5x\) with respect to \(x\) is:

\[
\int \cos 5x \, dx = \frac{1}{5} \sin 5x
\]

### Step 3: Combine the results
Combining both integrals, we get:

\[
\int (\sin 2x + \cos 5x) \, dx = -\frac{1}{2} \cos 2x + \frac{1}{5} \sin 5x + C
\]

Now, let's match this result with the given options:

The correct answer is:

\[
\boxed{-\frac{1}{2} \cos 2x + \frac{1}{5} \sin 5x + C}
\]

which corresponds to option **d** in the image.

Would you like more details on the integration steps, or do you have any questions?

Here are some related questions you might find interesting:

1. How do you integrate functions involving other trigonometric functions like \(\tan(x)\) or \(\sec(x)\)?
2. What are some common techniques for integrating products of trigonometric functions?
3. How do you apply integration by parts to trigonometric functions?
4. What are the applications of integration in physics, such as calculating energy or work?
5. How can you verify your integration results using differentiation?

**Tip:** When integrating trigonometric functions with coefficients, always account for the coefficient adjustment in your antiderivative.

¿Cuál es el resultado de la integral ∫(sen2x + cos5x)dx?

Learn how to solve the integral ∫(sin2x + cos5x)dx step-by-step. This problem demonstrates the integration of trigonometric functions, including sin and cos with coefficients. Suitable for students at the high school or early university level.

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