7.4.4How can I write simpler fractions?

Integration by Partial Fractions

7-210.

Evaluate $\int \frac { 8 x + 19 } { 2 x ^ { 2 } + 13 x + 15 } d x$ using the fact that $\frac { 8 x + 19 } { 2 x ^ { 2 } + 13 x + 15 } = \frac { 2 } { 2 x + 3 } + \frac { 3 } { x + 5 }$ .

7-211.

What are the values of $a$ and $b$ if $a(x-3)+b(x-7)=$ $3x-1$ ?

7-212.

PARTIAL FRACTIONS

Evaluating integrals with fractions such as $\int \frac { 8 x + 19 } { 2 x ^ { 2 } + 13 x + 15 } d x$ is made possible by rewriting the fraction as the sum of simpler fractions. These are called partial fractions because each simpler fraction is part of the sum.

To rewrite a complicated fraction such as the one above, first decide the form of the new fractions. Since adding fractions requires a least common denominator, the denominator of each partial fraction will be a factor of the denominator of the original fraction. Since $2x^2+13x+15=$ $(2x+3)(x+5)$ , the partial fractions will have the form $\frac { a } { 2 x + 3 }$ and $\frac { b } { x + 5 }$ , where $a$ and $b$ are values that we can solve for.

Solve for the values of $a$ and $b$ so that $\frac{8x+19}{2x^2+13x+15}=$ $\frac{a}{2x+3}+\frac{b}{x+5}$ is true for all values of $x$ .
Verify that the values for $a$ and $b$ result in the same partial fractions given in problem 7-210.

7-213.

Consider the function $f(x) =\frac { x + 16 } { x ^ { 2 } + 2 x - 8 }$ .

Rewrite $\frac { x + 16 } { x ^ { 2 } + 2 x - 8 }$ as the sum of partial fractions.
Use the partial fractions to integrate $\int \frac { x + 16 } { x ^ { 2 } + 2 x - 8 }dx$ .
Differentiate your answer to part (b) and simplify to verify that your antiderivative is correct

7-214.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method.

$\int x e ^ { - x } d x$

$\int \frac { x + 5 } { x ^ { 2 } - 1 } d x$

$\int \frac { 2 x d x } { x ^ { 2 } + 1 }$

$\int \frac { 2 d x } { x ^ { 2 } + 1 }$

$\int \frac { x ^ { 2 } + x - 1 } { x ( x ^ { 2 } - 1 ) } d x$

7-215.

Compute without a calculator No calculator! Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int \operatorname { cos } ^ { - 1 } ( x ) d x$
$\int _ { \pi / 3 } ^ { \pi / 2 } \operatorname { csc } ( x ) \operatorname { cot } ( x ) d x$
$\int _ { - 1 } ^ { 3 } \frac { d x } { x ^ { 2 / 3 } }$
$\int \operatorname { sec } ^ { 2 } ( x ) \operatorname { tan } ( x ) d x$

7-216.

Yvette claims that $f(x) = x^{1/3}$ has no points of inflection because $f^{\prime\prime}(x)$ is never equal to zero. Do you agree or disagree? If you disagree, explain the error in Yvette’s reasoning. Homework Help ✎

7-217.

Consider the function $f(x) = x^{4/3} - 4x^{1/3}$ . Homework Help ✎

Algebraically determine where $f$ is increasing.
Algebraically determine whether $f$ is concave up or concave down at $x = -1$ .
Determine the $x$ -coordinates of all points of inflection.

7-218.

In a hotly contested tug-of-war, the center of the rope moves back and forth according to the equation $x(t)=$ $\frac{3}{2}-\cos(t)-\frac{1}{2}\cos(2t)$ , where $x$ is measured in feet. Homework Help ✎

Calculate the displacement of the center of the rope over the interval $0 ≤ t ≤ π$ .
Calculate the total distance traveled by the center of the rope over the interval $0 ≤ t ≤ π$ .

7-219.

A particle moves along the curve $y = 2^x$ so that its $x$ -coordinate increases at the constant rate of $5$ units per second. How fast is its angle of inclination $θ$ changing when $x = 2$ ?

(As shown in the diagram at right, $θ$ is the angle between the segment joining the point to the origin and the $x$ -axis.) Homework Help ✎

Increasing exponential curve, with point on the curve in first quadrant, segment connects the point to the origin, angle between segment and positive x axis, labeled theta.

7-220.

Evaluate each limit. Homework Help ✎

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { cos } ( 3 x ) - \operatorname { cos } ( x ) } { x ^ { 2 } }$

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { tan } ( x ) } { e ^ { x } - 1 }$

$\lim\limits _ { x \rightarrow 0 ^ { + } } ( \operatorname { cot } ( x ) - \operatorname { csc } ( x ) )$

$\lim\limits _ { x \rightarrow 0 } \frac { 2 ^ { x } } { x }$

7-221.

Multiple Choice: $\int \operatorname { cos } \sqrt { x } d x =$ Homework Help ✎

$\sin\sqrt { x }+C$

$2x \sin(x) - 2\sin(x) + C$

$2\sqrt { x }\sin \sqrt { x }+ 2 \cos\sqrt { x }+C$

$2x \cos(x) - 2\sin(x) + C$

$2\sqrt{x}\cos\sqrt{x} + 2 \cos\sqrt{x} + C$

Calculus Third Edition