Let's solve the given integral step by step. The integral is:

$\int 8x^5 \cos(4x^6) \, dx$

Step 1: Substitution

Let $u = 4x^6$. Then, differentiate $u$ with respect to $x$: $\frac{du}{dx} = 24x^5 \quad \text{or} \quad du = 24x^5 \, dx$

From this, solve for $x^5 \, dx$: $x^5 \, dx = \frac{1}{24} du$

Step 2: Rewrite the Integral

Substitute $u = 4x^6$ and $x^5 \, dx = \frac{1}{24} du$ into the integral: $\int 8x^5 \cos(4x^6) \, dx = \int 8 \cdot \frac{1}{24} \cos(u) \, du$

Simplify the constants: $\int 8x^5 \cos(4x^6) \, dx = \frac{8}{24} \int \cos(u) \, du = \frac{1}{3} \int \cos(u) \, du$

Step 3: Integrate

The integral of $\cos(u)$ is $\sin(u)$: $\frac{1}{3} \int \cos(u) \, du = \frac{1}{3} \sin(u) + C$

Step 4: Substitute Back $u = 4x^6$

Replace $u$ with $4x^6$ to return to the original variable: $\frac{1}{3} \sin(u) + C = \frac{1}{3} \sin(4x^6) + C$

Final Answer:

$\int 8x^5 \cos(4x^6) \, dx = \frac{1}{3} \sin(4x^6) + C$

Would you like further clarifications or detailed explanations of any step?

Related Questions:

1. What is the general method of substitution in integration?
2. How do you handle integrals with trigonometric functions?
3. Can substitution work for definite integrals? How?
4. What are other strategies for solving integrals that cannot use substitution?
5. How does the chain rule in differentiation relate to substitution in integration?

Tip:

When solving integrals using substitution, always verify by differentiating your result to check if it matches the original integrand.