Antiderivatives (Indefinite Integrals) 1

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Question 1 of 5

1. Question
Integrate

$\int x^{3} d x$ $int x^3 dx$
- 1. $\frac{x^{4}}{3} + 5$ $x^4/3 +5$
- 2. $x^{4} + c$ $x^4 + c$
- 3. $\frac{x^{4}}{3} + c$ $x^4/3 + c$
- 4. $\frac{x^{4}}{4} + c$ $x^4/4 + c$
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Integration Formula

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

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

$=$ $=$ $\frac{x^{4}}{4} + c$ $(x^(4))/(4) +c$

$\frac{x^{4}}{4} + c$ $(x^(4))/(4) +c$

Question 2 of 5

Integrate

\int (5 - 2 x - x^{2}) d x

$int (5-2x-x^2)dx$

1. $x^{2} - \frac{x^{3}}{3} + c$ $x^2 -x^3/3 + c$
2. $- x^{2} - \frac{x^{3}}{3} + c$ $-x^2 -x^3/3 + c$
3. $x - x^{2} - \frac{x^{3}}{3} + c$ $x -x^2 -x^3/3 + c$
4. $5 x - x^{2} - \frac{x^{3}}{3} + c$ $5x -x^2 -x^3/3 + c$

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Integration Formula

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + c

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

Sum or Difference Rule

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

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

Apply the Sum or Difference Rule

\int (5 - 2 x - x^{2}) d x

$int (5-2x-x^2) dx$

=

$=$

\int 5 d x - \int 2 x d x - \int x^{2} d x

$int 5 dx-int 2x dx-int x^2 dx$

Find the Indefinite Integral

$\int 5 d x - \int 2 x d x - \int x^{2} d x$ $int5dx-int2xdx-intx^2dx$	$=$ $=$	$\int5x^{\color{#004ec4}{0}}dx-\int2x^{\color{#004ec4}{1}}dx-\int x^{\color{#004ec4}{2}}dx$	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

	$=$ $=$	$5 \frac{x^{1}}{1} - 2 \frac{x^{2}}{2} - \frac{x^{3}}{3} + c$ $5x^1/1-2x^2/2-x^3/3+c$
	$=$ $=$	$5 x - x^{2} - \frac{x^{3}}{3} + c$ $5x-x^2-x^3/3+c$

5 x - x^{2} - \frac{x^{3}}{3} + c

$5x-x^2-x^3/3+c$

Question 3 of 5

Integrate

\int {(1 - 2 x)}^{6} d x

$int(1-2x)^6dx$

1. ${(1 - 2 x)}^{7} + c$ $(1-2x)^7+c$
2. $\frac{1}{5} {(1 - 2 x)}^{5} + c$ $1/5 (1-2x)^5+c$
3. $\frac{1}{7} {(1 - 2 x)}^{7} + c$ $1/7 (1-2x)^7+c$
4. $- \frac{1}{14} {(1 - 2 x)}^{7} + c$ $-1/14 (1-2x)^7+c$

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Integration Formula

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

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

Integrate

$int(5x-4)^5dx$

1. $5/6 (5x-4)^6+c$
2. $1/30 (5x-4)^6+c$
3. $5 (5x-4)^4+c$
4. $1/6 (5x-4)^6+c$

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Integration Formula

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

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

Integrate

$int sqrt{x}(1+1/sqrt{x})dx$

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

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Integration Formula

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

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

	$=$	$\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{x^{1}}{1}+c$

	$=$	$\frac{2}{3}\sqrt{x^3}+x+c$

$2/3 sqrt{x^3}+x+c$

Antiderivatives (Indefinite Integrals) 1

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Quiz summary

Information

1. Question

Hint

Nice Job!

Integration Formula

2. Question

Hint

Excellent!

Integration Formula

Sum or Difference Rule

3. Question

Hint

Correct!

Integration Formula

4. Question

Hint

Great Work!

Integration Formula

5. Question

Hint

Great Work!

Integration Formula

Sum or Difference Rule

Quizzes

$\int x^{\color{#004ec4}{3}} dx$	$=$ $=$	$\frac{x^{\color{#004ec4}{3}+1}}{\color{#004ec4}{3}+1}+c$	Apply the standard formula

	$=$ $=$	$\frac{x^{4}}{4} + c$ $(x^(4))/(4) +c$