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The integral you provided is:

\[
\int t^7 \sin(2t^4) \, dt
\]

### Solution:

We will solve this using substitution. 

Let:
\[
u = t^4
\]

Now, differentiate \(u\) with respect to \(t\):
\[
du = 4t^3 \, dt \quad \text{or} \quad dt = \frac{du}{4t^3}
\]

Substituting \(u = t^4\) into the integral:

The integral becomes:
\[
\int t^7 \sin(2t^4) \, dt = \int t^7 \sin(2u) \

Evaluate the following integral: ∫ t^7 sin(2t^4) dt

This page provides a detailed step-by-step solution to the integral ∫ t^7 sin(2t^4) dt using substitution. The problem involves substitution and fundamental trigonometric integral techniques. Perfect for students studying calculus at an advanced high school or college level.

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