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Antiderivatives (Indefinite Integrals) 1Antiderivatives (Indefinite Integrals) 1
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Question 1 of 5
1. Question
Integrate`int x^3 dx`Hint
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Integration Formula
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$Find the Indefinite Integral$$\int x^{\color{#004ec4}{3}} dx$$ `=` $$\frac{x^{\color{#004ec4}{3}+1}}{\color{#004ec4}{3}+1}+c$$ Apply the standard formula `=` `(x^(4))/(4) +c` `(x^(4))/(4) +c ` -
Question 2 of 5
2. Question
Integrate`int (5-2x-x^2)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
$$ \int (f(x) \pm g(x))dx = \int f(x)dx \pm \int g(x)dx = F(x) \pm G(x) + c $$Apply the Sum or Difference Rule`int (5-2x-x^2) dx` `=` `int 5 dx-int 2x dx-int x^2 dx` Find the Indefinite Integral`int5dx-int2xdx-intx^2dx` `=` $$\int5x^{\color{#004ec4}{0}}dx-\int2x^{\color{#004ec4}{1}}dx-\int x^{\color{#004ec4}{2}}dx$$ Take the constants out of the integral signs `=` $$5\int x^{\color{#004ec4}{0}}dx-2\int x^{\color{#004ec4}{1}}dx-\int x^{\color{#004ec4}{2}}dx$$ `=` $$5\frac{x^{\color{#004ec4}{0}+1}}{\color{#004ec4}{0}+1}-2\frac{x^{\color{#004ec4}{1}+1}}{\color{#004ec4}{1}+1}-\frac{x^{\color{#004ec4}{2}+1}}{\color{#004ec4}{2}+1}+c$$ Use the Integration Formula `=` `5x^1/1-2x^2/2-x^3/3+c` `=` `5x-x^2-x^3/3+c` `5x-x^2-x^3/3+c` -
Question 3 of 5
3. Question
Integrate`int(1-2x)^6dx`Hint
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Integration Formula
$$\int f(x)^{\color{#004ec4}{n}} dx=\frac{f(x)^{\color{#004ec4}{n}+1}}{(\color{#004ec4}{n}+1)f'(x)}+c$$Find the Indefinite Integral$$\int (1-2x)^{\color{#004ec4}{6}}dx$$ `=` $$\frac{(1-2x)^{\color{#004ec4}{6}+1}}{(\color{#004ec4}{6}+1)(1-2x)’}+c$$ Apply the Integration Formula `=` $$\frac{(1-2x)^{\color{#004ec4}{7}}}{\color{#004ec4}{7}\times(-2)}+c$$ `=` `-1/14 (1-2x)^7+c` `-1/14 (1-2x)^7+c` -
Question 4 of 5
4. Question
Integrate`int(5x-4)^5dx`Hint
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Integration Formula
$$\int f(x)^{\color{#004ec4}{n}} dx=\frac{f(x)^{\color{#004ec4}{n}+1}}{(\color{#004ec4}{n}+1)f'(x)}+c$$Find the Indefinite Integral$$\int (5x-4)^{\color{#004ec4}{5}}dx$$ `=` $$\frac{(5x-4)^{\color{#004ec4}{5}+1}}{(\color{#004ec4}{5}+1)(5x-4)’}+c$$ Apply the Integration Formula `=` $$\frac{(5x-4)^{\color{#004ec4}{6}}}{\color{#004ec4}{6}\times 5}+c$$ `=` `1/30 (5x-4)^6+c` `1/30 (5x-4)^6+c` -
Question 5 of 5
5. Question
Integrate`int sqrt{x}(1+1/sqrt{x})dx`Hint
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Integration Formula
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$Sum or Difference Rule
$$ \int (f(x) \pm g(x))dx = \int f(x)dx \pm \int g(x)dx = F(x) \pm G(x) + c $$`sqrt{x}(1+1/sqrt{x})` can be written as$$\color{#004ec4}{\sqrt{x}}(1+\frac{1}{\sqrt{x}})$$ `=` $$\color{#004ec4}{\sqrt{x}}\times 1+\color{#004ec4}{\sqrt{x}}\times\frac{1}{\sqrt{x}}$$ `=` $$\sqrt{x}+1$$ Apply the Sum or Difference Rule$$\int\sqrt{x}(1+\frac{1}{\sqrt{x}})dx$$ `=` $$\int(\sqrt{x}+1)dx$$ `=` $$\int\sqrt{x}dx +\int 1dx$$ Find the Indefinite Integral$$\int\sqrt{x}dx +\int 1dx$$ `=` $$\int x^\color{#004ec4}{\frac{1}{2}}dx +\int 1x^\color{#004ec4}{0}dx$$ An alternative way to write the equation `=` $$\frac{x^{\color{#004ec4}{\frac{1}{2}}+1}}{\color{#004ec4}{\frac{1}{2}}+1}+\frac{x^{\color{#004ec4}{0}+1}}{\color{#004ec4}{0}+1}+c$$ Apply the Integration Formula `=` $$\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{x^{1}}{1}+c$$ `=` $$\frac{2}{3}\sqrt{x^3}+x+c$$ `2/3 sqrt{x^3}+x+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