Calculus for Team-Based Inquiry Learning: 2022 Edition

Activity 5.6.1.

Using the method of substitution, which of these is the integral of ${\displaystyle \int \frac{1}{(ax+b)}dx\text{?}}$

1. ${\displaystyle \frac{1}{a}\ln (ax+b)+C}$

2. ${\displaystyle \frac{1}{a}\ln |ax+b|+C}$

3. ${\displaystyle \frac{1}{x}(\ln (ax)+\ln (b))+C}$

4. ${\displaystyle \frac{1}{a}(\ln |ax|+\ln |b|)+C}$

Activity 5.6.2.

Which of the following is equivalent to the rational expression ${\displaystyle \frac{3x+3}{2x^2+3x}\text{?}}$

1. ${\displaystyle \frac{3}{2x}+\frac{1}{x}}$

2. ${\displaystyle \frac{3}{x}+\frac{3}{2x+3}}$

3. ${\displaystyle \frac{1}{x}+\frac{1}{2x+3}}$

4. ${\displaystyle \frac{3}{2x^2}+1}$

Activity 5.6.3.

Based on your choice in Activity 5.6.2, which of these is the integral for ${\displaystyle \int \frac{3x+3}{2x^2+3x}dx\text{?}}$

1. ${\displaystyle \frac{3}{2}\ln |x|+\ln |x|+C}$

2. ${\displaystyle \ln |x|+\frac{1}{2}\ln |2x+3|+C}$

3. ${\displaystyle -\frac{3}{2}{x}^{-1}+x+C}$

4. ${\displaystyle 3\ln |x|+\frac{3}{2}\ln |2x+3|+C}$

Activity 5.6.4.

Which of the following is an antiderivative for ${\displaystyle \frac{1}{(x+k{)}^2+h^2}\text{?}}$

1. ${\displaystyle \arctan (x)}$

2. ${\displaystyle \arctan (x+k)}$

3. ${\displaystyle \frac{1}{h}\arctan \left(\frac{x+h}{h}\right)}$

4. ${\displaystyle \frac{1}{h^2}\arctan \left(\frac{x+k}{h^2}\right)}$

5. ${\displaystyle \frac{1}{h}\arctan \left(\frac{x+k}{h}\right)}$

Activity 5.6.5.

Consider the irreducible quadratic $x^2-4x+80\text{.}$ Which of the following is the completed square form of this quadratic expression?

1. ${\displaystyle (x-4{)}^2+80}$

2. ${\displaystyle (x-2{)}^2+80}$

3. ${\displaystyle (x-4{)}^2+64}$

4. ${\displaystyle (x-2{)}^2+76}$

5. ${\displaystyle (x-4{)}^2+76}$

Activity 5.6.6.

Using the fact that ${\displaystyle \int \frac{1}{x^2-8x+80}dx=\int \frac{1}{(x-4{)}^2+64}dx\text{,}}$ evaluate the integral ${\displaystyle \int \frac{1}{x^2-8x+80}dx\text{.}}$

Activity 5.6.7.

Suppose that one could write ${\displaystyle \frac{4x^2-5x-2}{x^2(x-2)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-2}\text{.}}$

Recall that:

What are the values of $A,B$ and $C\text{?}$

1. ${\displaystyle A=4,B=-5,C=-2}$

2. ${\displaystyle A=3,B=1,C=1}$

3. ${\displaystyle A=1,B=1,C=1}$

4. ${\displaystyle A=1,B=3,C=-1}$

5. ${\displaystyle A=2,B=1,C=1}$

Activity 5.6.9.

Which of the following is the form of the partial fraction decomposition of ${\displaystyle \frac{x^3-7x^2-7x+15}{x^3(x+5)}\text{?}}$

1. ${\displaystyle \frac{A}{x}+\frac{B}{x+5}}$

2. ${\displaystyle \frac{A}{x^3}+\frac{B}{x+5}}$

3. ${\displaystyle \frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{D}{x+5}}$

4. ${\displaystyle \frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{Dx+E}{x+5}}$

Activity 5.6.10.

Which of the following is the form of the partial fraction decomposition of ${\displaystyle \frac{x^2+1}{(x-3{)}^2(x^2+4{)}^2}\text{?}}$

1. ${\displaystyle \frac{A}{x-3}+\frac{B}{(x-3{)}^2}+\frac{C}{x^2+4}+\frac{D}{(x^2+4{)}^2}}$

2. ${\displaystyle \frac{A}{x-3}+\frac{B}{(x-3{)}^2}+\frac{Cx+D}{(x^2+4{)}^2}}$

3. ${\displaystyle \frac{A}{x-3}+\frac{B}{(x-3{)}^2}+\frac{C}{x^2+4}+\frac{Dx+E}{(x^2+4{)}^2}}$

4. ${\displaystyle \frac{A}{x-3}+\frac{B}{(x-3{)}^2}+\frac{Cx+D}{x^2+4}+\frac{Ex+F}{(x^2+4{)}^2}}$

Activity 5.6.11.

Consider that the partial decomposition of ${\displaystyle \frac{x^2+5x+3}{(x+1{)}^{2}x}}$ is

What equality do we obtain if we multiply both sides of the above equation by $(x+1{)}^{2}x\text{?}$

1. ${\displaystyle x^2+5x+3=Ax(x+1)+Bx+C(x+1{)}^2}$

2. ${\displaystyle x^2+5x+3=A(x+1)+B(x+1{)}^2+Cx}$

3. ${\displaystyle x^2+5x+3=Ax(x+1)+Bx+C(x+1)}$

4. ${\displaystyle x^2+5x+3=A(x+1)+Bx+C{x}^2}$

5. ${\displaystyle x^2+5x+3=Ax(x+1)+B{x}^2+C(x+1{)}^2}$

Activity 5.6.12.

Notice that $x^2+5x+3=Ax(x+1)+Bx+C(x+1{)}^2\text{.}$

Activity 5.6.13.

Rewrite $Ax(x+1)+x+3(x+1{)}^2$ so that $Ax(x+1)+x+3(x+1{)}^2=?x^2+?x+?\text{.}$ Knowing that

can we solve for $A\text{?}$

Activity 5.6.14.

By using the form of the decomposition ${\displaystyle \frac{x^2+5x+3}{(x+1{)}^{2}x}=\frac{A}{x+1}+\frac{B}{(x+1{)}^2}+\frac{C}{x}}$ and the coefficients found in Activity 5.6.12 and Activity 5.6.13, evaluate ${\displaystyle \int \frac{x^2+5x+3}{(x+1{)}^{2}x}dx\text{.}}$

Activity 5.6.15.

Given that ${\displaystyle \frac{x^3-7x^2-7x+15}{x^3(x+5)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{D}{x+5}\text{,}}$ what are $A,B,C\text{,}$ and $D\text{?}$

1. ${\displaystyle A=1,B=-7C=-7,D=15}$

2. ${\displaystyle A=-1,B=-2,C=3,D=2}$

3. ${\displaystyle A=1,B=-2,C=-3,D=2}$

4. ${\displaystyle A=-1,B=2,C=3,D=-2}$

Activity 5.6.16.

Given your choice in Activity 5.6.15 Find ${\displaystyle \int \frac{x^3-7x^2-7x+15}{x^3(x+5)}dx.}$

Activity 5.6.17.

Consider the rational expression ${\displaystyle \frac{2x^3+2x+4}{x^4+2x^3+4x^2}.}$ Which of the following is the partial fraction decomposition of this rational expression?

1. ${\displaystyle \frac{1}{x}+\frac{1}{x^2}+\frac{2x-1}{x^2+2x+4}}$

2. ${\displaystyle \frac{2}{x}+\frac{0}{x^2}+\frac{-1}{x^2+2x+4}}$

3. ${\displaystyle \frac{0}{x}+\frac{1}{x^2}+\frac{-1}{x^2+2x+4}}$

4. ${\displaystyle \frac{0}{x}+\frac{1}{x^2}+\frac{2x-1}{x^2+2x+4}}$

Activity 5.6.18.

Given your choice in Activity 5.6.17 Find ${\displaystyle \int \frac{2x^3+2x+4}{x^4+2x^3+4x^2}dx\text{.}}$

Activity 5.6.19.

Given that ${\displaystyle \frac{2x+5}{x^2+3x+2}=\frac{-1}{x+2}+\frac{3}{x+1}\text{,}}$ find ${\displaystyle {\int }_0^3\frac{2x+5}{x^2+3x+2}dx\text{.}}$

Activity 5.6.21.

Evaluate ${\displaystyle \int \frac{4x^2-3x+1}{(2x+1)(x+2)(x-3)}dx\text{.}}$