To solve the given assertion and reason, we will analyze both statements step by step.
### Step 1: Understand the Assertion
The assertion states that:
\[
\int e^x [\sin x - \cos x] \, dx = e^x \sin x + C
\]
### Step 2: Analyze the Integral
We can rewrite the integral:
\[
\int e^x [\sin x - \cos x] \, dx = \int e^x \sin x \, dx - \int e^x \cos x \, dx
\]
### Step 3: Apply Integration by Parts
We will use integration by parts for both integrals. Recall the formula:
\[
\int u \, dv = uv - \int v \, du
\]
Let’s first compute \(\int e^x \sin x \, dx\):
- Let \(u = \sin x\) and \(dv = e^x \, dx\)
- Then, \(du = \cos x \, dx\) and \(v = e^x\)
Applying integration by parts:
\[
\int e^x \sin x \, dx = e^x \sin x - \int e^x \cos x \, dx
\]
Now we need to compute \(\int e^x \cos x \, dx\):
- Let \(u = \cos x\) and \(dv = e^x \, dx\)
- Then, \(du = -\sin x \, dx\) and \(v = e^x\)
Applying integration by parts again:
\[
\int e^x \cos x \, dx = e^x \cos x + \int e^x \sin x \, dx
\]
### Step 4: Set Up the Equation
Now we have two equations:
1. \(\int e^x \sin x \, dx = e^x \sin x - \int e^x \cos x \, dx\)
2. \(\int e^x \cos x \, dx = e^x \cos x + \int e^x \sin x \, dx\)
Let \(I_1 = \int e^x \sin x \, dx\) and \(I_2 = \int e^x \cos x \, dx\). We can express \(I_1\) and \(I_2\) in terms of each other:
\[
I_1 = e^x \sin x - I_2
\]
\[
I_2 = e^x \cos x + I_1
\]
### Step 5: Substitute and Solve
Substituting \(I_2\) into the equation for \(I_1\):
\[
I_1 = e^x \sin x - (e^x \cos x + I_1)
\]
\[
I_1 + I_1 = e^x \sin x - e^x \cos x
\]
\[
2I_1 = e^x (\sin x - \cos x)
\]
\[
I_1 = \frac{1}{2} e^x (\sin x - \cos x)
\]
### Step 6: Conclusion
Thus, we find:
\[
\int e^x [\sin x - \cos x] \, dx = \frac{1}{2} e^x (\sin x - \cos x) + C
\]
This shows that the assertion is incorrect because it states the integral equals \(e^x \sin x + C\).
### Step 7: Analyze the Reason
The reason states:
\[
\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C
\]
This is a correct statement, as it follows from the product rule of differentiation.
### Final Conclusion
- The assertion (A) is **false**.
- The reason (R) is **true**.
