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Indefinite Integrals of Logarithmic Functions 2Indefinite Integrals of Logarithmic Functions 2
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Question 1 of 5
1. Question
Find the integral$$\int{\frac{3}{2x-6}} dx$$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.$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$ `=` $$\int \frac{\color{#9a00c7}{3}}{\color{#D800AD}{2x-6}}$$ `=` `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$$\int{\frac{x^2}{2x^3+4}} dx$$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.$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$ `=` $$\int \frac{\color{#9a00c7}{x^2}}{\color{#D800AD}{2x^3+4}}$$ `=` `(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$$\int{\frac{x^5}{2x^6+3}} dx$$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.$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$ `=` $$\int \frac{\color{#9a00c7}{x^5}}{\color{#D800AD}{2x^6+3}}$$ `=` `(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$$\int{\frac{x}{(x+3)(x-3)}} dx$$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.$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$ `=` $$\int \frac{\color{#9a00c7}{x}}{\color{#D800AD}{(x+3)(x-3)}}$$ `=` $$\frac{x}{\color{#CC0000}{x^2-9}}$$ 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$$\int{\frac{x+3}{x^2+6x-1}} dx$$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.$$\int \frac{\color{#9a00c7}{f'(x)}}{\color{#D800AD}{f(x)}}$$ `=` $$\int \frac{\color{#9a00c7}{x+3}}{\color{#D800AD}{x^2+6x-1}}$$ `=` `(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
- Indefinite Integrals 1
- Indefinite Integrals 2
- Indefinite Integrals 3
- Indefinite Integrals of Exponential Functions
- Indefinite Integrals of Logarithmic Functions 1
- Indefinite Integrals of Logarithmic Functions 2
- Indefinite Integrals of Trig Functions
- Definite Integrals
- Definite Integrals of Exponential Functions
- Definite Integrals of Logarithmic Functions
- Definite Integrals of Trig Functions
- Areas Between Curves and the Axis 1
- Areas Between Curves and the Axis 2
- Area Between Curves
- Volumes of Revolution 1
- Volumes of Revolution 2
- Volumes of Revolution 3
- Trapezoidal Rule
- Simpsons Rule
- Applications of Integration for Trig Functions