Antiderivatives of Logarithmic Functions 2

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Question 1 of 5

1. Question

Find the integral

$$\int{\frac{3}{2x-6}} dx$$

`3/2 ln (2x-6)+c`
`2 ln (2x-6)+c`
`3 ln (2x-6)+c`
`3/2 ln (2x)+c`

Hint

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Correct

Nice Job!

Incorrect

Indefinite Integral

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx=\log_e [\color{#D800AD}{f(x)}]+c$$

First, form a fraction to balance the equation.

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$	`=`	$$\int \frac{\color{#9a00c7}{3}}{\color{#D800AD}{2x-6}}$$

	`=`	`3/2`	Differentiate the denominator

Use `3/2` as a constant to balance the integral.

`f(x)`	`=`	`3x-6`
`f'(x)`	`=`	`3`

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx$$	`=`	$$\log_e [\color{#D800AD}{f(x)}]+c$$

$$\int \frac{\color{#9a00c7}{3}}{\color{#D800AD}{2x-6}}dx$$	`=`	$$\frac{3}{2} \ln [\color{#D800AD}{2x-6}]+c$$	Substitute known values

`3/2 ln (2x-6)+c`

Question 2 of 5

2. Question

Find the integral

$$\int{\frac{x^2}{2x^3+4}} dx$$

`1/6 ln (2x^3+4)+c`
`1/2 ln (2x^3+4)+c`
`ln (2x^3+4)+c`
`1/3 ln (2x^3+4)+c`

Hint

Help Video

Correct

Well Done!

Incorrect

Indefinite Integral

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx=\log_e [\color{#D800AD}{f(x)}]+c$$

First, form a fraction to balance the equation.

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$	`=`	$$\int \frac{\color{#9a00c7}{x^2}}{\color{#D800AD}{2x^3+4}}$$

	`=`	`(x^2)/(6x^2)`	Differentiate the denominator

	`=`	`1/6`	`(x^2)/(x^2)=1`

Use `1/6` as a constant to balance the integral.

`f(x)`	`=`	`2x^3+4`
`f'(x)`	`=`	`6x^2`

$$\frac{1}{6}\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx$$	`=`	$$\frac{1}{6}\log_e [\color{#D800AD}{f(x)}]+c$$

$$\frac{1}{6}\int \frac{\color{#9a00c7}{6x^2}}{\color{#D800AD}{2x^3+4}}dx$$	`=`	$$\frac{1}{6} \ln [\color{#D800AD}{2x^3+4}]+c$$	Substitute known values

`1/6 ln (2x^3+4)+c`

Question 3 of 5

3. Question

Find the integral

$$\int{\frac{x^5}{2x^6+3}} dx$$

`1/12 ln (2x^6+3)+c`
`1/6 ln (2x^6+3)+c`
`12 ln (2x^6+3)+c`
`ln (2x^6+3)+c`

Hint

Help Video

Correct

Correct!

Incorrect

Indefinite Integral

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx=\log_e [\color{#D800AD}{f(x)}]+c$$

First, form a fraction to balance the equation.

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$	`=`	$$\int \frac{\color{#9a00c7}{x^5}}{\color{#D800AD}{2x^6+3}}$$

	`=`	`(x^5)/(12x^5)`	Differentiate the denominator

	`=`	`1/12`	`(x^5)/(x^5)=1`

Use `1/12` as a constant to balance the integral.

`f(x)`	`=`	`2x^6+3`
`f'(x)`	`=`	`12x^5`

$$\frac{1}{12}\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx$$	`=`	$$\frac{1}{12}\log_e [\color{#D800AD}{f(x)}]+c$$

$$\frac{1}{12}\int \frac{\color{#9a00c7}{12x^5}}{\color{#D800AD}{2x^6+3}}dx$$	`=`	$$\frac{1}{12} \ln [\color{#D800AD}{2x^6+3}]+c$$	Substitute known values

`1/12 ln (2x^6+3)+c`

Question 4 of 5

4. Question

Find the integral

$$\int{\frac{x}{(x+3)(x-3)}} dx$$

`1/2 ln (x^2-9)+c`
`1/2 ln (x-3)+c`
`2 ln (x^2-9)+c`
`1/2 ln (x+3)+c`

Hint

Help Video

Correct

Fantastic!

Incorrect

Indefinite Integral

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx=\log_e [\color{#D800AD}{f(x)}]+c$$

First, form a fraction to balance the equation.

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$	`=`	$$\int \frac{\color{#9a00c7}{x}}{\color{#D800AD}{(x+3)(x-3)}}$$

	`=`	$$\frac{x}{\color{#CC0000}{x^2-9}}$$	Expand

	`=`	`x/(2x)`	Differentiate the denominator

	`=`	`1/2`	`(x)/(x)=1`

Use `1/2` as a constant to balance the integral.

`f(x)`	`=`	`x^2-9`
`f'(x)`	`=`	`2x`

$$\frac{1}{2}\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx$$	`=`	$$\frac{1}{2}\log_e [\color{#D800AD}{f(x)}]+c$$

$$\frac{1}{2}\int \frac{\color{#9a00c7}{2x}}{\color{#D800AD}{x^2-9}}dx$$	`=`	$$\frac{1}{2} \ln [\color{#D800AD}{x^2-9}]+c$$	Substitute known values

`1/2 ln (x^2-9)+c`

Question 5 of 5

5. Question

Find the integral

$$\int{\frac{x+3}{x^2+6x-1}} dx$$

`1/2 ln (x^2+6x-1)+c`
`1/2 ln (x^2-6x-1)+c`
`1/2 ln (6x-1)+c`
`ln (x^2+6x-1)+c`

Hint

Help Video

Correct

Nice Job!

Incorrect

Indefinite Integral

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx=\log_e [\color{#D800AD}{f(x)}]+c$$

First, form a fraction to balance the equation.

$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$	`=`	$$\int \frac{\color{#9a00c7}{x+3}}{\color{#D800AD}{x^2+6x-1}}$$

	`=`	`(x+3)/(2x+6)`	Differentiate the denominator

	`=`	$$\frac{x+3}{\color{#CC0000}{2(x+3)}}$$	Factorize

	`=`	`1/2`	`(x+3)/(x+3)=1`

Use `1/2` as a constant to balance the integral.

`f(x)`	`=`	`x^2+6x-1`
`f'(x)`	`=`	`2x+6`

$$\frac{1}{2}\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx$$	`=`	$$\frac{1}{2}\log_e [\color{#D800AD}{f(x)}]+c$$

$$\frac{1}{2}\int \frac{\color{#9a00c7}{2x+6}}{\color{#D800AD}{x^2+6x-1}}dx$$	`=`	$$\frac{1}{2} \ln [\color{#D800AD}{x^2+6x-1}]+c$$	Substitute known values

`1/2 ln (x^2+6x-1)+c`

Quizzes

Antiderivatives (Indefinite Integrals) 1
Antiderivatives (Indefinite Integrals) 2
Antiderivatives (Indefinite Integrals) 3
Antiderivatives of Exponential Functions
Antiderivatives of Logarithmic Functions 1
Antiderivatives of Logarithmic Functions 2
Antiderivatives of Trig Functions 1
Antiderivatives of Trig Functions 2
Definite Integrals
Definite Integrals of Exponential Functions
Definite Integrals of Logarithmic Functions