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Antiderivatives (Indefinite Integrals) 2Antiderivatives (Indefinite Integrals) 2
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Question 1 of 5
1. Question
Integrate`int (3x-6)^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
$$ \int (f(x) \pm g(x))dx = \int f(x)dx \pm \int g(x)dx = F(x) \pm G(x) + c $$Formula of Reduced Multiplication (the squared difference)
$$(a-b)^2= a^2-2ab+b^2$$Apply the Formula of Reduced Multiplication to the expression `(3x-6)^2``(3x-6)^2` `=` $$(3x)^2-2\times3x\times6+6^2$$ `=` $$9x^2-36x+36$$ Apply the Sum or Difference Rule$$\int (3x-6)^2dx$$ `=` $$\int(9x^2-36x+36)dx$$ `=` $$\int 9x^2dx-\int 36xdx+\int 36dx $$ Find the Indefinite Integral$$\int 9x^2dx-\int 36xdx+\int 36dx $$ Take the constants out of the integral signs `=` $$9\int x^2dx-36\int xdx+36\int 1dx $$ `=` $$9\int x^\color{#004ec4}{2}dx-36\int x^\color{#004ec4}{1}dx+36\int x^\color{#004ec4}{0}dx $$ `=` $$9\frac{x^{\color{#004ec4}{2}+1}}{\color{#004ec4}{2}+1}-36\frac{x^{\color{#004ec4}{1}+1}}{\color{#004ec4}{1}+1}+36\frac{x^{\color{#004ec4}{0}+1}}{\color{#004ec4}{0}+1}+c$$ Apply the Integration Formula `=` $$9\frac{x^{3}}{3}-36\frac{x^{2}}{2}+36\frac{x^{1}}{1}+c$$ `=` $$3x^{3}-18x^{2}+36x+c$$ `3x^{3}-18x^{2}+36x+c` -
Question 2 of 5
2. Question
Integrate`int sqrt(3x-5) 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 sqrt{3x-5} dx ` can be written as `int (3x-5)^{1/2}dx`Find the Indefinite Integral$$\int{(3x-5)}^{\color{#004ec4}{\frac{1}{2}}}dx$$ `=` $$\frac{(3x-5)^{\color{#004ec4}{\frac{1}{2}}+1}}{(\color{#004ec4}{\frac{1}{2}}+1)(3x-5)’}+c$$ Apply the Integration Formula `=` $$\frac{(3x-5)^{\color{#004ec4}{\frac{3}{2}}}}{\color{#004ec4}{\frac{3}{2}}\times 3}+c$$ `=` `2/9 (3x-5)^{3/2}+c` `=` `2/9 sqrt{(3x-5)^3}+c` `{2sqrt{(3x-5)^3}}/9 + c` -
Question 3 of 5
3. Question
Integrate$$\int\frac{6}{\sqrt[3]{x}}dx$$Hint
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Integration Formula
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$Find the Indefinite Integral$$\int\frac{6}{\sqrt[3]{x}}dx$$ `=` $$6\int\frac{1}{\sqrt[3]{x}}dx$$ Take the constant `6` out of the integral sign `=` $$6\int x^{\color{#004ec4}{-\frac{1}{3}}}dx$$ The integral can be written `=` $$6\frac{x^{\color{#004ec4}{-\frac{1}{3}}+1}}{\color{#004ec4}{-\frac{1}{3}}+1}+c$$ Apply the Integration Formula `=` $$ 6\frac{x^{\color{#004ec4}{\frac{2}{3}}}}{\color{#004ec4}{\frac{2}{3}}}+c$$ `=` $$\frac{18}{2} x^{ \frac{2}{3} }+c$$ `=` `9x^{2/3} + c` `9x^{2/3} + c` -
Question 4 of 5
4. Question
Integrate$$\int\sqrt[3]{4x+3}$$Hint
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Integration Formula
$$\int f(x)^{\color{#004ec4}{n}} dx=\frac{f(x)^{\color{#004ec4}{n}+1}}{(\color{#004ec4}{n}+1)f'(x)}+c$$$$\int\sqrt[3]{4x+3} dx $$ can be written as $$\int(4x+3)^{\frac{1}{3}}dx$$Find the Indefinite Integral$$\int{(4x+3)}^{\color{#004ec4}{\frac{1}{3}}}dx$$ `=` $$\frac{(4x+3)^{\color{#004ec4}{\frac{1}{3}}+1}}{(\color{#004ec4}{\frac{1}{3}}+1)(4x+3)’}+c$$ Apply the Integration Formula `=` $$\frac{(4x+3)^{\color{#004ec4}{\frac{4}{3}}}}{\color{#004ec4}{\frac{4}{3}}\times 4}+c$$ `=` $$\frac{(4x+3)^{{\frac{4}{3}}}}{{\frac{16}{3}}}+c$$ `=` $$\frac{3}{16}(4x+3)^{\frac{4}{3}}+c$$ `3/16 (4x+3)^{4/3}+c` -
Question 5 of 5
5. Question
Integrate`int 1/{3(4x-5)^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 1/{3(4x-5)^3}dx` can be written as `int 1/3 (4x-5)^{-3}dx`Find the Indefinite Integral$$\int\frac{1}{3}(4x-5)^{\color{#004ec4}{-3}}dx$$ `=` $$\frac{1}{3}\int(4x-5)^{\color{#004ec4}{-3}}dx$$ Take the constant `1/3` out of the integral sign `=` $$\frac{1}{3}\frac{(4x-5)^{\color{#004ec4}{-3}+1}}{(\color{#004ec4}{-3}+1)(4x-5)’}+c$$ Apply the Integration Formula `=` $$\frac{1}{3}\frac{(4x-5)^{\color{#004ec4}{-2}}}{\color{#004ec4}{-2}\times 4}+c$$ `=` $$\frac{1}{3}(-\frac{1}{8})(4x-5)^{-2}+c$$ `=` $$-\frac{1}{24}(4x-5)^{-2}+c$$ Simplify `=` $$-\frac{1}{24(4x-5)^2}+c$$ Apply Negative Indice law $$-\frac{1}{24(4x-5)^2}+c$$
Quizzes
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- Antiderivatives (Indefinite Integrals) 3
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- Antiderivatives of Logarithmic Functions 1
- Antiderivatives of Logarithmic Functions 2
- Antiderivatives of Trig Functions 1
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- Definite Integrals
- Definite Integrals of Exponential Functions
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