Evaluate the following integrals using the appropriate rules of integration. Show all your calculations in each case.
a.

Ask by Valdez Ruiz. in South Africa

Mar 28,2025

Question

Evaluate the following integrals using the appropriate rules of integration. Show all your calculations in each case. a. $$ \int \frac{t^{2}+8}{t^{2}-5 t+6} d t $$

UpStudy AI · Accepted Answer

We start with the integral

$$
\int \frac{t^2 + 8}{t^2 - 5t + 6}\,dt.
$$

**Step 1. Factor and Divide**

Notice that the denominator factors as

$$
t^2 - 5t + 6 = (t-2)(t-3).
$$

Since the degrees of the numerator and denominator are equal, we perform polynomial long division. Dividing $$t^2 + 8$$ by $$t^2 - 5t + 6$$, we get:

$$
t^2 + 8 = \bigl(t^2 - 5t + 6\bigr)\cdot 1 + \bigl(5t+2\bigr).
$$

Thus,

$$
\frac{t^2 + 8}{t^2 - 5t + 6} = 1 + \frac{5t+2}{t^2-5t+6}.
$$

So the integral becomes

$$
\int \frac{t^2 + 8}{t^2 - 5t + 6}\,dt = \int 1\,dt + \int \frac{5t+2}{(t-2)(t-3)}\,dt.
$$

**Step 2. Partial Fraction Decomposition**

We decompose

$$
\frac{5t+2}{(t-2)(t-3)}
$$

as

$$
\frac{5t+2}{(t-2)(t-3)} = \frac{A}{t-2} + \frac{B}{t-3}.
$$

Multiply both sides by $$(t-2)(t-3)$$:

$$
5t+2 = A(t-3) + B(t-2).
$$

Expand the right-hand side:

$$
5t+2 = At - 3A + Bt - 2B = (A+B)t - (3A+2B).
$$

Equate the coefficients of like terms:

For $$t$$:
$$
A + B = 5,
$$
and for the constant term:
$$
- (3A + 2B) = 2 \quad \Longrightarrow \quad 3A + 2B = -2.
$$

Solve the system:
- From $$A + B = 5$$, express $$A$$ as $$A = 5 - B$$.
- Substitute into $$3A + 2B = -2$$:

$$
3(5-B) + 2B = -2,
$$
$$
15 - 3B + 2B = -2,
$$
$$
15 - B = -2,
$$
$$
-B = -17,
$$
$$
B = 17.
$$

Then,

$$
A = 5 - 17 = -12.
$$

Thus, the partial fractions are

$$
\frac{5t+2}{(t-2)(t-3)} = \frac{-12}{t-2} + \frac{17}{t-3}.
$$

**Step 3. Integrate Each Term**

The integral now is:

$$
\int 1\,dt + \int \left(\frac{-12}{t-2} + \frac{17}{t-3}\right)\,dt.
$$

Integrate term by term:

1. For the constant:
   $$
   \int 1\,dt = t.
   $$
2. For the first partial fraction:
   $$
   \int \frac{-12}{t-2}\,dt = -12 \ln |t-2|.
   $$
3. For the second partial fraction:
   $$
   \int \frac{17}{t-3}\,dt = 17 \ln |t-3|.
   $$

**Final Answer**

Combining these results, we obtain:

$$
\int \frac{t^2 + 8}{t^2 - 5t + 6}\,dt = t - 12\ln|t-2| + 17\ln|t-3| + C,
$$

where $$C$$ is the constant of integration.
$$ \int \frac{t^2 + 8}{t^2 - 5t + 6}\,dt = t - 12\ln|t-2| + 17\ln|t-3| + C $$

Evaluate the following integrals using the appropriate rules of integration. Show all your calculations in each case.
a.

Upstudy AI Solution

Answer

Solution

Mind Expander

Step 1: Polynomial Long Division

Step 2: Rewrite the Integral

Step 3: Evaluate the First Integral

Step 4: Evaluate the Second Integral Using Partial Fraction Decomposition

Step 5: Set Up Equations

Step 6: Rewrite the Integral Using These Constants

Step 7: Evaluate the Integrals

Step 8: Combine All Parts

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