1. Perform the division x^2 + 3 / (x^2 - 7x + 10) and find the integral from 6 to 7 of (x^2 + 3) / (x^2 - 7x + 10) dx. 2. Explain how to calculate d/dx (ln(sec(x))) using the chain... 1. Perform the division x^2 + 3 / (x^2 - 7x + 10) and find the integral from 6 to 7 of (x^2 + 3) / (x^2 - 7x + 10) dx. 2. Explain how to calculate d/dx (ln(sec(x))) using the chain rule. 3. Find the integral from π/4 to π/2 of dx/sin(x). 4. Put z = (2 - 5j) / (3 + 4j) into complex exponential form re^(iθ).

Understand the Problem

The question is asking to solve various mathematical problems related to calculus and complex numbers, demonstrating knowledge of integration, differentiation, and conversions to exponential form.

Answer

The integral evaluates to: $$ 1 + \frac{28}{3} \ln(2) - \frac{7}{3} \ln\left(\frac{5}{4}\right). $$

Steps to Solve

Divide the Polynomials To divide the polynomial $\frac{x^2 + 3}{x^2 - 7x + 10}$, use polynomial long division.

Divide the leading term: $x^2$ by $x^2$ gives $1$.
Multiply the divisor by $1$: $(x^2 - 7x + 10) \cdot 1 = x^2 - 7x + 10$.
Subtract from the original numerator: $$ (x^2 + 3) - (x^2 - 7x + 10) = 7x - 7. $$

So we can write: $$ \frac{x^2 + 3}{x^2 - 7x + 10} = 1 + \frac{7x - 7}{x^2 - 7x + 10} $$

Partial Fraction Decomposition Now express $\frac{7x - 7}{x^2 - 7x + 10}$ using partial fractions. Factor the denominator if possible.

The quadratic $x^2 - 7x + 10$ factors to $(x - 5)(x - 2)$.

Therefore, $$ \frac{7x - 7}{(x - 5)(x - 2)} = \frac{A}{x - 5} + \frac{B}{x - 2} $$

Multiply through by the denominator $(x - 5)(x - 2)$ to find $A$ and $B$: $$ 7x - 7 = A(x - 2) + B(x - 5). $$

Set Up System of Equations Expand and group terms: $$ 7x - 7 = (A + B)x + (-2A - 5B). $$

Equate coefficients:

$A + B = 7$
$-2A - 5B = -7$

Solve for A and B From $A + B = 7$, we can express $A$ in terms of $B$: $A = 7 - B$.

Substituting into the second equation: $$ -2(7 - B) - 5B = -7 $$ $$ -14 + 2B - 5B = -7 $$ $$ -3B = 7 $$ $$ B = -\frac{7}{3}, \quad A = 7 - B = 7 + \frac{7}{3} = \frac{28}{3}. $$

So the partial fraction decomposition is: $$ \frac{7x - 7}{(x - 5)(x - 2)} = \frac{28/3}{x - 5} - \frac{7/3}{x - 2}. $$

Integrate the Expression Now we can find the integral: $$ \int_6^7 \left(1 + \frac{28/3}{x - 5} - \frac{7/3}{x - 2}\right) dx. $$

This breaks down into three parts: $$ \int_6^7 dx + \frac{28}{3} \int_6^7 \frac{1}{x - 5} dx - \frac{7}{3} \int_6^7 \frac{1}{x - 2} dx $$

Perform the Integrals Calculate each integral:

The first integral: $$ \int_6^7 dx =[x]_6^7 = 7 - 6 = 1. $$
The second integral: $$ \int_6^7 \frac{1}{x - 5} dx = [\ln|x - 5|]_6^7 = \ln|2| - \ln|1| = \ln(2). $$
The third integral: $$ \int_6^7 \frac{1}{x - 2} dx = [\ln|x - 2|]_6^7 = \ln(5) - \ln(4) = \ln\left(\frac{5}{4}\right). $$

Combining these: $$ \int_6^7 f(x) , dx = 1 + \frac{28}{3} \ln(2) - \frac{7}{3} \ln\left(\frac{5}{4}\right). $$

The final answer for the integral is: $$ 1 + \frac{28}{3} \ln(2) - \frac{7}{3} \ln\left(\frac{5}{4}\right). $$

More Information

This answer uses polynomial long division and partial fraction decomposition, which is a common technique in calculus for simplifying integrals involving rational functions.

Tips

Failing to correctly factor the quadratic equation in the denominator.
Not properly simplifying the expressions for $A$ and $B$ in partial fractions.
Incorrect limits when integrating; always check the bounds.

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