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Antiderivatives (Indefinite Integrals) 3Antiderivatives (Indefinite Integrals) 3
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Question 1 of 5
1. Question
Integrate`int {x^3+4} /x^3 dx`Hint
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Integration Formula
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$Sum or Difference Rule
$$ \int (f(x) \pm g(x))dx = \int f(x)dx \pm \int g(x)dx = F(x) \pm G(x) + c $$Addition of Fractions Rule
`(a+b)/c=a/c+b/c`Apply the Addition of Fractions Rule`int {x^3+4}/x^3dx` `=` `int(x^3/x^3+4/x^3)dx` `=` `int(1+4/x^3)dx` Use the Sum or Difference Rule`int(1+4/x^3)dx` `=` `int 1dx+int 4/x^3dx` Find the Indefinite Integral`int 1dx+int 4/x^3dx` `=` $$\int x^{\color{#004ec4}{0}}dx+\int 4x^{\color{#004ec4}{-3}}dx$$ Take the constant `4` out of the integral sign `=` $$\int x^{\color{#004ec4}{0}}dx+4\int x^{\color{#004ec4}{-3}}dx$$ `=` $$\frac{x^{\color{#004ec4}{0}+1}}{\color{#004ec4}{0}+1}+4\frac{x^{\color{#004ec4}{-3}+1}}{\color{#004ec4}{-3}+1}+c$$ Apply the Integration Formula `=` `x^1/1+4x^{-2}/{-2}+c` `=` $$x+4(-\frac{1}{2})x^{-2}+c$$ `=` `x-2x^{-2}+c` Simplify `=` `x-2/x^{2}+c` Apply Negative Indice law `x-2/x^2+c` -
Question 2 of 5
2. Question
Integrate`int3/(6-x)^4dx`Hint
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Integration Formula
$$\int f(x)^{\color{#004ec4}{n}} dx=\frac{f(x)^{\color{#004ec4}{n}+1}}{(\color{#004ec4}{n}+1)f'(x)}+c$$`int 3/(6-x)^4dx` can be written as `int 3(6-x)^{-4}dx`Find the Indefinite Integral$$\int 3(6-x)^{\color{#004ec4}{-4}}dx$$ `=` $$3\int(6-x)^{\color{#004ec4}{-4}}dx$$ Take the constant `3` out of the integral sign `=` $$3\frac{(6-x)^{\color{#004ec4}{-4}+1}}{(\color{#004ec4}{-4}+1)(6-x)’}+c$$ Apply the Integration Formula `=` $$3\frac{(6-x)^{\color{#004ec4}{-3}}}{(\color{#004ec4}{-3})(-1)}+c$$ `=` $$(6-x)^{-3}+c$$ Simplify `=` `1/(6-x)^3+c` Apply Negative Indice law `1/(6-x)^3+c` -
Question 3 of 5
3. Question
Integrate`int 1/(x^4) dx`Hint
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Integration Formula
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$Remove the fraction by expressing the term using negative exponents`int 1/x^4 dx` `=` $$\int x^{\color{#004ec4}{-4}} dx$$ Find the Indefinite Integral$$\int x^{\color{#004ec4}{-4}} dx$$ `=` $$\frac{x^{\color{#004ec4}{-4}+1}}{\color{#004ec4}{-4}+1}+c$$ Apply the Integration Formula `=` `(x^(-3))/(-3) +c` `=` `-1/(3x^3) + c` Apply Negative Indice law `-1/(3x^3) + c` -
Question 4 of 5
4. Question
Integrate`int 2/(3 sqrt(x)) dx`Hint
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Integration Formula
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$`int 2/{3sqrt{x}}dx` can be written as `int 2/3 x^{-1/2}dx`Find the Indefinite Integral$$\int{\frac{2}{3}x^{-\frac{1}{2}}}dx $$ `=` $$\frac{2}{3}\int{x^{-\frac{1}{2}}}dx$$ Take the constant `2/3` out of the integral sign `=` $$\frac{2}{3}\frac{x^{\color{#004ec4}{-\frac{1}{2}}+1}}{(\color{#004ec4}{-\frac{1}{2}}+1)}+c$$ Apply the Integration Formula `=` $$\frac{2}{3}\frac{x^{\color{#004ec4}{\frac{1}{2}}}}{\color{#004ec4}{\frac{1}{2}}}+c$$ `=` $$\frac{2}{3}(2)x^\frac{1}{2}+c$$ `=` $$\frac{4}{3}x^{\frac{1}{2}}+c$$ Simplify `=` $$\frac{4}{3}\sqrt{x}+c$$ Convert into surd form $$\frac{4}{3}\sqrt{x}+c$$ -
Question 5 of 5
5. Question
Integrate`int 3/sqrt(2x-3) dx`Hint
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Integration Formula
$$\int f(x)^{\color{#004ec4}{n}} dx=\frac{f(x)^{\color{#004ec4}{n}+1}}{(\color{#004ec4}{n}+1)f'(x)}+c$$`int 3/sqrt{2x-3}dx` can be written as `int 3(2x-3)^{-1/2}dx`Find the Indefinite Integral$$\int3(2x-3)^{\color{#004ec4}{-\frac{1}{2}}}dx$$ `=` $$3\int(2x-3)^{\color{#004ec4}{-\frac{1}{2}}}dx$$ Take the constant `3` out of the integral sign `=` $$3\frac{(2x-3)^{\color{#004ec4}{-\frac{1}{2}}+1}}{(\color{#004ec4}{-\frac{1}{2}}+1)(2x-3)’}+c$$ Apply the Integration Formula `=` $$3\frac{(2x-3)^{\color{#004ec4}{\frac{1}{2}}}}{\color{#004ec4}{\frac{1}{2}}\times 2}+c$$ `=` `3 (2x-3)^{1/2}+c` Simplify `=` `3 sqrt{2x-3}+c` Convert into surd form `3 sqrt{2x-3}+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