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Wize University Calculus 1 Textbook > Integrals

Properties of Definite Integrals

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Properties of the Definite Integral

  1. ∫aaf(x)dx=0\boxed{\displaystyle \int_{a}^{a}f(x)\text{d}x=0}
  2. ∫abkdx=k(b−a)\displaystyle \boxed{\int_{a}^{b}k\text{d}x=k(b-a)} for any constant kk
  3. ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx\displaystyle \boxed{\int_{a}^{b}f(x)\text{d}x=\int_{a}^{c}f(x)\text{d}x+\int_{c}^{b}f(x)\text{d}x} for all cc in [a, b][a,\ b]
  4. ∫ab[f(x)±g(x)]dx=∫abf(x)dx±∫abg(x)dx\boxed{\displaystyle \int_{a}^{b}[f(x)\pm g(x)]\text{d}x=\int_{a}^{b}f(x)\text{d}x\pm\int_{a}^{b}g(x)\text{d}x}
  5. ∫abkf(x)dx=k∫abf(x)dx\displaystyle \boxed{\int_{a}^{b}kf(x)\text{d}x=k\int_{a}^{b}f(x)\text{d}x} for any constant kk
  6. ∫abf(x)dx=−∫baf(x)dx\boxed{\displaystyle \int_{a}^{b}f(x)\text{d}x=-\int_{b}^{a}f(x)\text{d}x}
  7. ∫−aaf(x) dx=0,   if f(x) is an odd function\boxed{\displaystyle\int_{-a}^af\left(x\right)\text{ d}x=0 \text{, \ \ if } f(x) \text{ is an odd function}}
  8. ∫−aaf(x) dx=2∫0af(x) dx,   if f(x) is an even function\boxed{\displaystyle \int_{-a}^{a}f(x)\text{ d}x=2\int_0^af(x)\text{ d}x \text{, \ \ if } f(x) \text{ is an even function}}

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Mean Value Theorem for Definite Integrals

If f(x)f\left(x\right)is a continuous function on [a,b]\left[a,b\right]then there exists a≤c≤ba\le c\le b such that

∫abf(x)dx=f(c)(b−a)\boxed{\displaystyle \int^b_af(x)dx=f(c)(b-a)}

Suppose f(−x)=−f(x)f(-x)=-f(x) and g(−x)=g(x)g\left(-x\right)=g\left(x\right) for all xx values.

If ∫01f(x)dx=5  and  ∫−10g(x)dx=7\displaystyle\int_0^1f\left(x\right)dx=5\ \ \text{and}\ \ \int_{-1}^0g\left(x\right)dx=7, evaluate ∫−11[2f(x)+3g(x)]dx\displaystyle \int_{-1}^1\left[2f\left(x\right)+3g\left(x\right)\right]dx
Suppose f(x)f(x) is continuous and let ∫07f(x)dx=7\displaystyle \int_0^7f\left(x\right)dx=7 and ∫010f(x)dx=10\displaystyle \int_0^{10}f\left(x\right)dx=10.

Find 5∫710f(x)dx\displaystyle 5\int_{7}^{10}f\left(x\right)dx.

Extra Practice
Properties of definite integrals
If ∫08f(x)dx=5,∫03f(x)dx=4,\int_0^8f(x)dx=5,\int_0^3f(x)dx=4, and ∫48f(x)dx=−2\int_4^8f(x)dx=-2, then what is the value of
∫34f(x)dx\int_3^4f(x)dx?
Evaluate the definite integral ∫01x(5−x2)dx\int_0^1x(5-x^2)dx.
Evaluate the definite integral ∫01x(5−x2)dx\int_0^1x(5-x^2)dx.
Evaluate the definite integral ∫01x(5−x2)dx\int_0^1x(5-x^2)dx.
Find the definite integral ∫02x(x+1)dx\displaystyle\int_0^2x(x+1)dx.
Definite with Absolute Value
Evaluate the following integral: ∫03∣x2−x−2∣ dx\displaystyle\int_0^3|x^2-x-2| \ dx
Definite Integral: Trigonometric Function and Absolute Value
Evaluate the definite integral: ∫05π/4∣sin⁡x∣ dx\displaystyle\int_0^{5\pi/4} |\sin x| \ dx
Practice: Definite with Fraction
Q:\textbf{Q:} Evaluate ∫14[1x2−x]dx\displaystyle \int_1^4\left[\frac{1}{x^2}-x\right]dx
Practice: Definite with Absolue Value
Q.\textbf{Q.} Compute the definite integral ∫−22(2−∣x∣)dx\displaystyle \int_{-2}^{2}(2-\lvert x\rvert)\text{d}x
Definite Integrals: Integration by Substitution
If ∫−27g(x)dx=13\int_{-2}^7g\left(x\right)dx=13 and ∫572g(x)=10\int_5^72g\left(x\right)=10, then ∫−152g(2x)dx=\int_{-1}^{\frac{5}{2}}g\left(2x\right)dx=
Properties of Definite Integrals
Suppose that f(−x)=f(x)f\left(-x\right)=f\left(x\right) and g(−x)=−g(x)g\left(-x\right)=-g\left(x\right). If ∫−102f(x)=5\int_{-1}^02f\left(x\right)=5, the value of ∫−11[3f(x)+x⋅sin⁡x⋅g(x)]dx=\int_{-1}^1\left[3f\left(x\right)+x\cdot\sin x\cdot g\left(x\right)\right]dx=
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx
Find ∫0π∣cos⁡ x∣dx\int_0^{\pi}\left|\cos\ x\right|dx
Practice: Definite with Absolue Value
Q.\textbf{Q.} Compute the definite integral ∫−22(2−∣x∣)dx\displaystyle \int_{-2}^{2}(2-\lvert x\rvert)\text{d}x
Definite with Absolute Value
Evaluate the following integral: ∫03∣x2−x−2∣ dx\displaystyle\int_0^3|x^2-x-2| \ dx
Definite Integral: Trigonometric Function and Absolute Value
Evaluate the definite integral: ∫05π/4∣sin⁡x∣ dx\displaystyle\int_0^{5\pi/4} |\sin x| \ dx
Practice: Definite with Fraction
Q:\textbf{Q:} Evaluate ∫14[1x2−x]dx\displaystyle \int_1^4\left[\frac{1}{x^2}-x\right]dx
Properties of definite integrals
If ∫08f(x)dx=5,∫03f(x)dx=4,\int_0^8f(x)dx=5,\int_0^3f(x)dx=4, and ∫48f(x)dx=−2\int_4^8f(x)dx=-2, then what is the value of
∫34f(x)dx\int_3^4f(x)dx?
Find the value of the definite integral ∫02x(x+1)dx\displaystyle \int_0^2x(x+1)dx.
Find the definite integral ∫02x(x+1)dx\displaystyle\int_0^2x(x+1)dx.
Evaluate the definite integral ∫01x(5−x2)dx\int_0^1x(5-x^2)dx.
Antiderivatives: Trigonometric Functions
Evaluate ∫π3π∣cos⁡x∣dx\int_{\frac{\pi}{3}}^{\pi}\left|\cos x\right|dx.