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Antiderivatives of Logarithmic Functions 1Antiderivatives of Logarithmic Functions 1
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Question 1 of 5
1. Question
Find the integral$$\int{\frac{5}{x}} dx$$Hint
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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
2. Question
Find the integral$$\int3x^2+4x-{\frac{6}{x}} dx$$Hint
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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
3. Question
Find the integral$$\int{\frac{dx}{x+5}} dx$$Hint
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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
4. Question
Find the integral$$\int{\frac{2x}{x^2+1}} dx$$Hint
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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
5. Question
Find the integral$$\int{\frac{1}{4x-7}} dx$$Hint
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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`
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