Indefinite Integrals 2

Question 1 of 5

Integrate

$int (3x-6)^2 dx$

1. $3x^{3}+18x^{2}+36$
2. $3x^{3}-18x^{2}+c$
3. $3x^{3}+18x^{2}+36x+c$
4. $3x^{3}-18x^{2}+36x+c$

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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

$(3x-6)^2$

$=$

$(3x)^2-2\times3x\times6+6^2$

$=$

$9x^2-36x+36$

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

Integrate

$int sqrt(3x-5) dx$

1. ${9sqrt{(3x-5)^3}}/2 + c$
2. ${2sqrt{(3x-5)^3}}/9 + c$
3. $3/2sqrt{3x-5} + c$
4. ${2sqrt{(3x-5)^3}}/3 + c$

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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

Integrate

$\int\frac{6}{\sqrt[3]{x}}dx$

1. $9x^{3/2} + c$
2. $2x^{-2/3} + c$
3. $9x^{2/3} + c$
4. $1/9 x^{2/3} + c$

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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

Integrate

$\int\sqrt[3]{4x+3}$

1. $16/3 (4x+3)^{4/3}+c$
2. $3/16 (4x+3)^{4/3}+c$
3. $3 (4x+3)^{4/3}+c$
4. $3/16 (4x+3)^{3/4}+c$

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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

Integrate

$int 1/{3(4x-5)^3} dx$

1. $-1/{6(4x-5)^2} + c$
2. $-1/24 (4x-5)^2 + c$
3. $-1/{6(4x-5)}+ c$
4. $-1/{24(4x-5)^2} + c$

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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

$-\frac{1}{24(4x-5)^2}+c$

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Quiz summary

Information

1. Question

Hint

Correct!

Integration Formula

Sum or Difference Rule

Formula of Reduced Multiplication (the squared difference)

2. Question

Hint

Well Done!

Integration Formula

3. Question

Hint

Excellent!

Integration Formula

4. Question

Hint

Fantastic!

Integration Formula

5. Question

Hint

Nice Job!

Integration Formula

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