Let's break this down step by step.

We are given:

• $x = t^4$
• $dx = 4t^3 dt$

We want to simplify the integral $\int \frac{4t^3 \, dt}{(1+t)t^2}$ into a simpler form and then use the integration by parts formula: $\int u \, dv = uv - \int v \, du$

Step 1: Simplify the integral

We start with the integral: $I = \int \frac{4t^3 \, dt}{(1+t)t^2}$ We can simplify the integrand: $I = \int \frac{4t^3}{(1+t)t^2} \, dt = \int \frac{4t}{1+t} \, dt$

Step 2: Use the integration by parts formula

Now, we need to use the integration by parts formula. Let's choose $u$ and $dv$ from the integrand:

• Let $u = \frac{t}{1+t}$, so that $du = \frac{1}{(1+t)^2} \, dt$
• Let $dv = dt$, so that $v = t$

Now, applying the integration by parts formula: $\int u \, dv = uv - \int v \, du$

This gives: $I = \left[ \frac{t}{1+t} \cdot t \right] - \int t \cdot \frac{1}{(1+t)^2} \, dt$

Step 3: Simplify the remaining integral

Now, we need to simplify the second integral: $\int \frac{t}{(1+t)^2} \, dt$ This can be solved by substitution. Let $w = 1+t$, so $dw = dt$ and $t = w - 1$. Substituting, we get: $\int \frac{w-1}{w^2} \, dw = \int \frac{1}{w} \, dw - \int \frac{1}{w^2} \, dw$ Solving these integrals: $\int \frac{1}{w} \, dw = \ln |w| = \ln(1+t)$ $\int \frac{1}{w^2} \, dw = -\frac{1}{w} = -\frac{1}{1+t}$

Thus, the integral becomes: $\int \frac{t}{(1+t)^2} \, dt = \ln(1+t) + \frac{1}{1+t}$

Final Answer

Putting it all together: $I = \left[ \frac{t^2}{1+t} \right] - \left( \ln(1+t) + \frac{1}{1+t} \right) + C$ This is the simplified solution.

Do you need more details or clarification on any step?

Here are 5 related questions to explore further:

1. How do you perform integration by substitution?
2. What is the geometric interpretation of an integral by parts?
3. How can you simplify integrals involving rational functions?
4. How do you handle improper integrals or limits in integration?
5. Can you explain more about the substitution $w = 1+t$ and its justification?

Tip: Always check if a substitution or a change of variable can simplify the integral before attempting complex techniques like integration by parts!