Questions: Evaluate the integral: ∫(2 sin x + 3 cos x) dx

Evaluate the integral: ∫(2 sin x + 3 cos x) dx
Transcript text: Evaluate the integral: $\int(2 \sin x+3 \cos x) d x$
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Solution

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Solution Steps

To evaluate the integral of the function \(2 \sin x + 3 \cos x\), we can integrate each term separately. The integral of \(\sin x\) is \(-\cos x\), and the integral of \(\cos x\) is \(\sin x\). Therefore, the integral of \(2 \sin x\) is \(-2 \cos x\), and the integral of \(3 \cos x\) is \(3 \sin x\). Combine these results and add the constant of integration \(C\).

Step 1: Set Up the Integral

We need to evaluate the integral

\[ \int (2 \sin x + 3 \cos x) \, dx. \]

Step 2: Integrate Each Term

We can integrate each term separately:

  1. The integral of \(2 \sin x\) is

\[ -2 \cos x. \]

  1. The integral of \(3 \cos x\) is

\[ 3 \sin x. \]

Step 3: Combine the Results

Combining the results from the integrals, we have:

\[ \int (2 \sin x + 3 \cos x) \, dx = -2 \cos x + 3 \sin x + C, \]

where \(C\) is the constant of integration.

Final Answer

Thus, the final result of the integral is

\[ \boxed{-2 \cos x + 3 \sin x + C}. \]

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