Indefinite Integrals of Logarithmic Functions 1

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Question 1 of 5

1. Question
Find the integral

$\int{\frac{5}{x}} dx$
- 1. $5 ln 5x+c$
- 2. $5 ln x+c$
- 3. $ln x+c$
- 4. $ln 5x+c$
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Integral of a Fraction

$\int{\frac{1}{\color{#004ec4}{x}}}dx=\log_e \color{#004ec4}{x}+c$

Substitute the components into the formula

$\int{\frac{1}{\color{#004ec4}{x}}}dx$ $=$ $\log_e \color{#004ec4}{x}+c$

$\int{\frac{5}{\color{#004ec4}{x}}}dx$ $=$ $5\int{\frac{1}{\color{#004ec4}{x}}}dx$ Take out the constant $5$

$=$ $5\ln \color{#004ec4}{x}+c$ Substitute known values

$5 ln x+c$

Question 2 of 5

Find the integral

$\int3x^2+4x-{\frac{6}{x}} dx$

1. $x^3+2x^2-6 ln x+c$
2. $x^3-2x^2-6 ln x+c$
3. $6 ln x+c$
4. $x^3-2x^2+6 ln x+c$

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Integral of a Fraction

$\int{\frac{1}{\color{#004ec4}{x}}}dx=\log_e \color{#004ec4}{x}+c$

Substitute the components into the formula

$\int{\frac{1}{\color{#004ec4}{x}}}dx$

$=$

$\log_e \color{#004ec4}{x}+c$

	$\int 3x^2+4x-{\frac{6}{\color{#004ec4}{x}}}dx$

$=$	$\frac{3x^3}{3}+\frac{4x^2}{2}-6\int{\frac{1}{\color{#004ec4}{x}}}dx$	Take out the constants and integrate

$=$	$x^3+2x^2-6\ln x+c$	Simplify

$x^3+2x^2-6 ln x+c$

Question 3 of 5

Find the integral

$\int{\frac{dx}{x+5}} dx$

1. $5 lnx+c$
2. $5ln (x+5)+c$
3. $ln (x+5)+c$
4. $(x+5) ln (x+5)+c$

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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.

$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$	$=$	$\int \frac{\color{#9a00c7}{1}}{\color{#D800AD}{x+5}}$

	$=$	$1/1$	Differentiate the denominator

	$=$	$1$

Since it satisfies

$(f'(x))/(f(x))$ , the equation is already balanced.

Substitute the components to the formula

$f(x)$	$=$	$x+5$
$f'(x)$	$=$	$1$

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

$\int \frac{\color{#9a00c7}{dx}}{\color{#D800AD}{x+5}}dx$	$=$	$\ln [\color{#D800AD}{x+5}]+c$	Substitute known values

$ln (x+5)+c$

Question 4 of 5

Find the integral

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

1. $xln (x+1)+c$
2. $(x^2+1)ln (x^2+1)+c$
3. $ln (x^2+1)+c$
4. $x^2+1+c$

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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.

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

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

	$=$	$1$

Since it satisfies

$(f'(x))/(f(x))$ , the equation is already balanced.

Substitute the components to the formula

$f(x)$	$=$	$x^2+1$
$f'(x)$	$=$	$2x$

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

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

$ln (x^2+1)+c$

Question 5 of 5

Find the integral

$\int{\frac{1}{4x-7}} dx$

1. $1/4 ln (4x-7)+c$
2. $-7 ln (4x-7)+c$
3. $ln (4x-7)+c$
4. $4 ln (4x-7)+c$

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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.

$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$	$=$	$\int \frac{\color{#9a00c7}{1}}{\color{#D800AD}{4x-7}}$

	$=$	$1/4$	Differentiate the denominator

Use

$1/4$ as a constant to balance the integral.

$f(x)$	$=$	$4x-7$
$f'(x)$	$=$	$4$

$\frac{1}{4}\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}dx$	$=$	$\frac{1}{4}\log_e [\color{#D800AD}{f(x)}]+c$

$\frac{1}{4}\int \frac{\color{#9a00c7}{4}}{\color{#D800AD}{4x-7}}dx$	$=$	$\frac{1}{4}\int \frac{\color{#9a00c7}{4}}{\color{#D800AD}{4x-7}}$

	$=$	$\frac{1}{4} \ln [\color{#D800AD}{4x-7}]+c$	Substitute known values

$1/4 ln (4x-7)+c$

Indefinite Integrals of Logarithmic Functions 1

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Quiz summary

Information

1. Question

Hint

Great Work!

Integral of a Fraction

2. Question

Hint

Correct!

Integral of a Fraction

3. Question

Hint

Keep Going!

Indefinite Integral

4. Question

Hint

Fantastic!

Indefinite Integral

5. Question

Hint

Excellent!

Indefinite Integral

Quizzes

$\int{\frac{5}{\color{#004ec4}{x}}}dx$	$=$	$5\int{\frac{1}{\color{#004ec4}{x}}}dx$	Take out the constant $5$

	$=$	$5\ln \color{#004ec4}{x}+c$	Substitute known values