How do I integrate sin raise to the power 2 omega multiplied by t?

Integrating the expression \(\sin^2(\omega t)\) involves using a trigonometric identity to simplify the integral. The integral of \(\sin^2\) can be a bit tricky at first, but with the right approach, it becomes manageable. Let me guide you through the process step by step.

Applying the Trigonometric Identity

To tackle the integral of \(\sin^2(\omega t)\), we can use the well-known identity:

• \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\)

For our case, substituting \(x\) with \(\omega t\), we have:

• \(\sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2}\)

Setting Up the Integral

Now, we can rewrite the integral:

\(\int \sin^2(\omega t) \, dt = \int \frac{1 - \cos(2\omega t)}{2} \, dt\)

This simplifies to:

\(\frac{1}{2} \int (1 - \cos(2\omega t)) \, dt\)

Breaking Down the Integral

Next, we can split the integral into two parts:

• \(\frac{1}{2} \int 1 \, dt\)
• \(-\frac{1}{2} \int \cos(2\omega t) \, dt\)

Calculating Each Integral

Let's evaluate each integral separately:

• The first integral \(\int 1 \, dt = t\)
• The second integral \(\int \cos(2\omega t) \, dt = \frac{1}{2\omega} \sin(2\omega t)\), using the substitution method or knowing the integral of cosine.

Putting It All Together

Now substituting these results back into our expression, we have:

\(\frac{1}{2} \left(t - \frac{1}{2\omega} \sin(2\omega t)\right) + C\)

where \(C\) is the constant of integration. Simplifying this gives:

\(\frac{t}{2} - \frac{\sin(2\omega t)}{4\omega} + C\)

The Final Result

Thus, the integral of \(\sin^2(\omega t)\) is:

\(\int \sin^2(\omega t) \, dt = \frac{t}{2} - \frac{\sin(2\omega t)}{4\omega} + C\)

This process not only demonstrates the integration of \(\sin^2(\omega t)\) but also highlights the importance of using trigonometric identities to simplify complex integrals. If you have any more questions about integration or related topics, feel free to ask!