Definite Integrals
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Question 1 of 5
1. Question
Integrate$$\int_{1}^{4} (2x-2) dx$$Hint
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Integration Formula for Definite integral
$$\int_{\color{#9a00c7}{a}}^{\color{\green}{b}} f(x) dx=\left[F(x)\right]_{\color{#9a00c7}{a}}^{\color{\green}{b}}=F(\color{\green}{b})-F(\color{#9a00c7}{a})$$Integration Formula for Indefinite integral
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$Sum or Difference Rule
$$ \int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx = F(x) \pm G(x) + c $$Find the Indefinite Integral$$\int (2x-2) dx$$ `=` $$\int 2x dx-\int 2 dx$$ Use the Sum or Difference Rule `=` $$2\int x^{\color{#004ec4}{1}} dx-2\int x^{\color{#004ec4}{0}} dx$$ Take the constants out of the integral signs `=` $$2\frac{x^{\color{#004ec4}{1}+1}}{\color{#004ec4}{1}+1}+2\frac{x^{\color{#004ec4}{0}+1}}{\color{#004ec4}{0}+1}+c$$ Use the Integration Formula for Indefinite integral `=` `2x^2/2+2x^{1}/1+c` `=` `x^2-2x+c` Find the Definite Integral by using `F(x)=x^2-2x`$$\int_{\color{#9a00c7}{1}}^{\color{\green}{4}} (2x-2) dx$$ `=` $$\left[x^2-2x\right]_{\color{#9a00c7}{1}}^{\color{\green}{4}}$$ Use the Integration Formula for Definite integral to get the answer `=` `(\color{\green}{4}^2-2\times\color{\green}{4}) – (\color{#9a00c7}{1}^2- 2\times\color{#9a00c7}{1})` `=` `(16-8) – (1-2)` `=` `8 + 1` `=` `9` `9` -
Question 2 of 5
2. Question
Integrate$$\int_{0}^{5} 3x^{2} dx$$Hint
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Integration Formula for Definite integral
$$\int_{\color{#9a00c7}{a}}^{\color{\green}{b}} f(x) dx=\left[F(x)\right]_{\color{#9a00c7}{a}}^{\color{\green}{b}}=F(\color{\green}{b})-F(\color{#9a00c7}{a})$$Integration Formula for Indefinite integral
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$Find the Indefinite Integral$$\int 3x^{2} dx$$ `=` $$3\int x^{2} dx$$ Take the constant `3` out of the integral sign `=` $$3\frac{x^{\color{#004ec4}{2}+1}}{\color{#004ec4}{2}+1}+c$$ Use the Integration Formula for Indefinite integral `=` `3x^3/3+c` `=` `x^3+c` Find the Definite Integral by using `F(x)=x^3`$$\int_{\color{#9a00c7}{0}}^{\color{\green}{5}} 3x^{2} dx$$ `=` $$\left[x^3\right]_{\color{#9a00c7}{0}}^{\color{\green}{5}}$$ Use the Integration Formula for Definite integral to get the answer `=` `\color{\green}{5}^3 – \color{#9a00c7}{0}^3` `=` `5^{3}` `=` `125` `125` -
Question 3 of 5
3. Question
Integrate$$\int_{-2}^{2} x^{5} dx$$Hint
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Integration Formula for Definite integral
$$\int_{\color{#9a00c7}{a}}^{\color{\green}{b}} f(x) dx=\left[F(x)\right]_{\color{#9a00c7}{a}}^{\color{\green}{b}}=F(\color{\green}{b})-F(\color{#9a00c7}{a})$$Integration Formula for Indefinite integral
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$Find the Indefinite Integral$$\int x^{5} dx$$ `=` $$\frac{x^{\color{#004ec4}{5}+1}}{\color{#004ec4}{5}+1}+c$$ Use the Integration Formula for Indefinite integral `=` `x^6/6+c` Find the Definite Integral by using `F(x)=x^6/6`$$\int_{\color{#9a00c7}{-2}}^{\color{\green}{2}} x^{5} dx$$ `=` $$\left[\frac{x^6}{6}\right]_{\color{#9a00c7}{-2}}^{\color{\green}{2}}$$ Use the Integration Formula for Definite integral to get the answer `=` `\color{\green}{2}^6/6-\color{#9a00c7}{(-2)}^6/6` `=` `64/6-64/6` `=` `0` `0` -
Question 4 of 5
4. Question
Integrate$$\int_{1}^{2} (2x-1)^{2} dx$$Hint
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Integration Formula for Definite integral
$$\int_{\color{#9a00c7}{a}}^{\color{\green}{b}} f(x) dx=\left[F(x)\right]_{\color{#9a00c7}{a}}^{\color{\green}{b}}=F(\color{\green}{b})-F(\color{#9a00c7}{a})$$Integration Formula for Indefinite integral
$$\int f(x)^{\color{#004ec4}{n}} dx=\frac{f(x)^{\color{#004ec4}{n}+1}}{(\color{#004ec4}{n}+1)f'(x)}+c$$Find the Indefinite Integral$$\int (2x-1)^{2} dx$$ `=` $$\frac{(2x-1)^{\color{#004ec4}{2}+1}}{(\color{#004ec4}{2}+1)(2x-1)’}+c$$ Use the Integration Formula for Indefinite integral `=` $$\frac{(2x-1)^{3}}{3\times2}+c$$ `=` $$\frac{(2x-1)^{3}}{6}+c$$ Find the Definite Integral by using `F(x)=(2x-1)^{3}/6`$$\int_{\color{#9a00c7}{1}}^{\color{\green}{2}} (2x-1)^{2} dx$$ `=` $$\left[\frac{(2x-1)^{3}}{6}\right]_{\color{#9a00c7}{1}}^{\color{\green}{2}}$$ Use the Integration Formula for Definite integral to get the answer `=` `(2\times\color{\green}{2}-1)^3/6 – (2\times\color{#9a00c7}{1}-1)^3/6` `=` `3^3/6 – 1/6` `=` `26/6` `=` `4 1/3` `4 1/3` -
Question 5 of 5
5. Question
Integrate$$\int_{0}^{1} \frac{x^{3}-2x^{2}+4x}{x} dx$$Hint
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Integration Formula for Definite integral
$$\int_{\color{#9a00c7}{a}}^{\color{\green}{b}} f(x) dx=\left[F(x)\right]_{\color{#9a00c7}{a}}^{\color{\green}{b}}=F(\color{\green}{b})-F(\color{#9a00c7}{a})$$Integration Formula for Indefinite integral
$$\int x^{\color{#004ec4}{n}} dx=\frac{x^{\color{#004ec4}{n}+1}}{\color{#004ec4}{n}+1}+c$$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`Find the Indefinite Integral$$\int \frac{x^{3}-2x^{2}+4x}{x} dx$$ `=` $$\int (\frac{x^{3}}{x}-\frac{2x^{2}}{x}+\frac{4x}{x}) dx$$ Apply the Addition of Fractions Rule `=` $$\int (x^{2}-2x+4) dx$$ `=` $$\int x^{2} dx-\int 2x dx+\int 4 dx$$ Use the Sum or Difference Rule `=` $$\int x^{2} dx-2\int x dx+4\int x^{0} dx$$ Take the constants out of the integral signs `=` $$\frac{x^{\color{#004ec4}{2}+1}}{\color{#004ec4}{2}+1}-2\frac{x^{\color{#004ec4}{1}+1}}{\color{#004ec4}{1}+1}+4\frac{x^{\color{#004ec4}{0}+1}}{\color{#004ec4}{0}+1}+c$$ Use the Integration Formula for Indefinite integral `=` `x^3/3-2x^{2}/2+4x^{1}/1+c` `=` `x^3/3-x^{2}+4x+c` Find the Definite Integral by using `F(x)=x^3/3-x^{2}+4x`$$\int_{\color{#9a00c7}{0}}^{\color{\green}{1}} \frac{x^{3}-2x^{2}+4x}{x} dx$$ `=` $$\left[\frac{x^3}{3}-x^{2}+4x\right]_{\color{#9a00c7}{0}}^{\color{\green}{1}}$$ Use the Integration Formula for Definite integral `=` ` (\color{\green}{1}^3/3-\color{\green}{1}^{2}+4\times\color{\green}{1})-(\color{#9a00c7}{0}^3/3-\color{#9a00c7}{0}^{2}+4\times\color{#9a00c7}{0})` `=` `(1/3-1+4)-(0-0+0)` `=` `1/3+3` `=` `3 1/3` `3 1/3`
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