verify the identity using the subsituion t=a+bu int u^2bu/(a+bu)^2

Let's verify the substitution \( t = a + bu \) into the given expression:

\[
\frac{u^2 b u}{(a + b u)^2}
\]

### Step 1: Substitution \( t = a + bu \)
The substitution given is \( t = a + bu \). From this, we can solve for \( u \) as:

\[
u = \frac{t - a}{b}
\]

Now, substitute \( u = \frac{t - a}{b} \) into the expression \( \frac{u^2 b u}{(a + b u)^2} \).

### Step 2: Substituting \( u \) into the expression

First, calculate \( u^2 b u \):
\[
u^2 b u = \left( \frac{t - a}{b} \right)^2 b \cdot \frac{t - a}{b} = \frac{(t - a)^3}{b^2}
\]

Now, substitute \( t = a + bu \) into the denominator \( (a + b u)^2 \), which gives \( t^2 \):

\[
(a + b u)^2 = t^2
\]

### Step 3: Simplify the expression

Now, substitute these results back into the original expression:

\[
\frac{u^2 b u}{(a + b u)^2} = \frac{\frac{(t - a)^3}{b^2}}{t^2} = \frac{(t - a)^3}{b^2 t^2}
\]

### Final Answer:
So, after substituting \( t = a + bu \), the expression simplifies to:

\[
\frac{(t - a)^3}{b^2 t^2}
\]

Would you like more details or have any questions? Here are 5 related questions to expand on this:

1. How do you apply substitution in integrals using similar transformations?
2. What is the role of chain rule when substituting in derivatives?
3. How can you check if a substitution is valid in more complex expressions?
4. What does substitution tell us about the nature of variables in equations?
5. How does substitution help in solving differential equations?

**Tip:** Always remember to adjust both the numerator and denominator when making substitutions in rational expressions.

verify the identity using the substitution t = a + bu in the expression u^2bu/(a + bu)^2

Explore the verification of the identity using the substitution t = a + bu in the expression u^2bu/(a + bu)^2. This detailed solution simplifies the problem into (t - a)^3 / (b^2 t^2), demonstrating algebraic manipulation and substitution techniques. Suitable for Grades 10-12.

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