Indefinite Integrals 3

Question 1 of 5

Integrate

$int {x^3+4} /x^3 dx$

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

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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

$\int (f(x) \pm g(x))dx = \int f(x)dx \pm \int g(x)dx = F(x) \pm G(x) + c$

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

Integrate

$int3/(6-x)^4dx$

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

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Integration Formula

$\int f(x)^{\color{#004ec4}{n}} dx=\frac{f(x)^{\color{#004ec4}{n}+1}}{(\color{#004ec4}{n}+1)f'(x)}+c$

$int 3/(6-x)^4dx$ 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

Integrate

$int 1/(x^4) dx$

1. $-1/(3x^3) + c$
2. $x^3+c$
3. $x^3/3 + c$
4. $-1/(4x^4) + c$

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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

Integrate

$int 2/(3 sqrt(x)) dx$

1. $1/(3x^(2)) + c$
2. $x^2/3 + c$
3. $4/3 sqrt(x) + c$
4. $1/3 sqrt(x) + c$

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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

Integrate

$int 3/sqrt(2x-3) dx$

1. $3/sqrt{2x-3}+c$
2. $3 sqrt{2x-3}+c$
3. $6 sqrt{2x-3}+c$
4. $sqrt{2x-3}+c$

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Integration Formula

$\int f(x)^{\color{#004ec4}{n}} dx=\frac{f(x)^{\color{#004ec4}{n}+1}}{(\color{#004ec4}{n}+1)f'(x)}+c$

$int 3/sqrt{2x-3}dx$ 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

$3 sqrt{2x-3}+c$

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Quiz summary

Information

1. Question

Hint

Fantastic!

Integration Formula

Sum or Difference Rule

Addition of Fractions Rule

2. Question

Hint

Keep Going!

Integration Formula

3. Question

Hint

Exceptional!

Integration Formula

4. Question

Hint

Well Done!

Integration Formula

5. Question

Hint

Keep Going!

Integration Formula

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