tell me sin^3x integration,

im getting cosx+(cos^3x)/3

To find the integral of \(\sin^3x\), let's go through the correct process:

### Step 1: Express \(\sin^3x\) in terms of simpler functions.
Use the identity:
\[
\sin^3x = \sin x \cdot \sin^2x = \sin x \cdot (1 - \cos^2x)
\]
Thus,
\[
\sin^3x = \sin x - \sin x \cdot \cos^2x
\]

### Step 2: Integrate the expression.
Now, integrate term by term:
\[
\int \sin^3x \, dx = \int \sin x \, dx - \int \sin x \cdot \cos^2x \, dx
\]

#### Integrating \(\int \sin x \, dx\):
\[
\int \sin x \, dx = -\cos x
\]

#### Integrating \(\int \sin x \cdot \cos^2x \, dx\):
For this, use substitution. Let \(u = \cos x\), hence \(du = -\sin x \, dx\).

The integral becomes:
\[
\int \sin x \cdot \cos^2x \, dx = -\int u^2 \, du = -\frac{u^3}{3} = -\frac{\cos^3x}{3}
\]

### Step 3: Combine the results.
Putting it all together:
\[
\int \sin^3x \, dx = -\cos x + \frac{\cos^3x}{3} + C
\]
where \(C\) is the constant of integration.

### Conclusion:
The correct integral is:
\[
\boxed{-\cos x + \frac{\cos^3x}{3} + C}
\]

You mentioned that you were getting \( \cos x + \frac{\cos^3x}{3} \), but there should be a negative sign in front of \(\cos x\) in the final answer.

Would you like more details or have any questions? Here are some related questions you might explore:

1. How would you integrate \(\cos^3x\)?
2. What is the integral of \(\sin^nx\) for any integer \(n\)?
3. How does integration by parts apply to trigonometric functions?
4. Can we use reduction formulas for \(\sin^nx\) integration?
5. What are some common mistakes when integrating powers of trigonometric functions?

**Tip:** Always double-check the signs in trigonometric integrals, as they can easily affect the final result.

Discover how to integrate sin^3x correctly with detailed steps and explanations. Learn about trigonometric identities and integration techniques. Suitable for advanced high school students and anyone interested in trigonometric integrals.

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