To solve the integral \(\int t \sin(t^2) \cos^2(t^2) \, dt\), we can use a substitution method.

### Step 1: Substitution
Let \( u = t^2 \), then \( du = 2t \, dt \) or \( \frac{du}{2} = t \, dt \).

### Step 2: Rewrite the Integral
Substituting \( u = t^2 \) into the integral:

\[
\int t \sin(t^2) \cos^2(t^2) \, dt = \int \sin(u) \cos^2(u) \cdot \frac{du}{2}
\]

This simplifies to:

\[
\frac{1}{2} \int \sin(u) \cos^2(u) \, du
\]

### Step 3: Use a Trigonometric Identity
We can simplify the integrand further using the identity \( \cos^2(u) = \frac{1 + \cos(2u)}{2} \):

\[
\frac{1}{2} \int \sin(u) \cdot \frac{1 + \cos(2u)}{2} \, du
\]

This expands to:

\[
\frac{1}{4} \int \sin(u) \, du + \frac{1}{4} \int \sin(u) \cos(2u) \, du
\]

### Step 4: Solve Each Integral

1. The first integral \( \frac{1}{4} \int \sin(u) \, du \) is straightforward:

   \[
   \frac{1}{4} (-\cos(u)) = -\frac{1}{4} \cos(u)
   \]

2. The second integral \( \frac{1}{4} \int \sin(u) \cos(2u) \, du \) can be solved using integration by parts or recognizing it as a standard integral form:

   Let \( I = \int \sin(u) \cos(2u) \, du \). This can be evaluated using trigonometric identities, but let's recognize the pattern. After simplification:

   \[
   I = \frac{1}{2} \left(\frac{\cos(u)}{2} - \frac{\cos(3u)}{6}\right)
   \]

   Therefore:

   \[
   \frac{1}{4} I = \frac{1}{8} \cos(u) - \frac{1}{24} \cos(3u)
   \]

### Step 5: Combine and Substitute Back
Putting it all together:

\[
\int t \sin(t^2) \cos^2(t^2) \, dt = -\frac{1}{4} \cos(t^2) + \frac{1}{8} \cos(t^2) - \frac{1}{24} \cos(3t^2) + C
\]

Finally:

\[
\int t \sin(t^2) \cos^2(t^2) \, dt = \frac{1}{8} \cos(t^2) - \frac{1}{24} \cos(3t^2) + C
\]

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

Would you like to explore any of the steps in more detail or have any questions?

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Here are 5 related questions:

1. How would you integrate \(\int t \cos(t^2) \sin^2(t^2) \, dt\)?
2. What is the integral of \(\int t^3 \sin(t^4) \cos^4(t^4) \, dt\)?
3. How can you apply substitution to solve \(\int t \sin(t^2) e^{t^2} \, dt\)?
4. Can you solve \(\int \sin(x) \cos(x) \, dx\) using trigonometric identities?
5. What is the integral of \(\int \sin^2(u) \cos^2(u) \, du\)?

**Tip:** When dealing with trigonometric integrals, consider using identities like \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \) to simplify the integrand.

Solve the integral t sin(t^2) cos^2(t^2) dt step-by-step using substitution and trigonometric identities. Learn how to simplify the integrand using trigonometric identities and apply integration by parts. This problem is suitable for students in Grades 11-12.

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