# Properties of Definite Integration

### Properties of Definite Integration:

#### 1. Linearity Property:

• Statement: If $f\left(x\right)$ and $g\left(x\right)$ are integrable functions and $c$ is a constant, then ${\int }_{a}^{b}\left[c\cdot f\left(x\right)+g\left(x\right)\right]\text{\hspace{0.17em}}dx=c\cdot {\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx+{\int }_{a}^{b}g\left(x\right)\text{\hspace{0.17em}}dx$
• This property allows you to split an integral over the sum or difference of functions into separate integrals.

• Statement: If $f\left(x\right)$ is integrable on $\left[a,b\right]$ and $c$ is a constant, then ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx={\int }_{a}^{c}f\left(x\right)\text{\hspace{0.17em}}dx+{\int }_{c}^{b}f\left(x\right)\text{\hspace{0.17em}}dx$
• This property allows the integral over an interval to be split into two integrals over sub-intervals.

#### 3. Constant Multiple Property:

• Statement: If $f\left(x\right)$ is integrable on $\left[a,b\right]$ and $c$ is a constant, then ${\int }_{a}^{b}c\cdot f\left(x\right)\text{\hspace{0.17em}}dx=c\cdot {\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx$
• This property allows a constant factor to be pulled out of the integral.

#### 4. Symmetry Property:

• Statement: If $f\left(x\right)$ is integrable on $\left[-a,a\right]$, then ${\int }_{-a}^{a}f\left(x\right)\text{\hspace{0.17em}}dx=0$
• This property is helpful when dealing with functions that exhibit symmetry around the y-axis.

#### 5. Reversal of Limits Property:

• Statement: If $f\left(x\right)$ is integrable on $\left[a,b\right]$, then ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx=-{\int }_{b}^{a}f\left(x\right)\text{\hspace{0.17em}}dx$
• This property allows you to switch the limits of integration while introducing a negative sign.

#### 6. Integral of a Constant:

• Statement: For any constant $c$, ${\int }_{a}^{b}c\text{\hspace{0.17em}}dx=c\cdot \left(b-a\right)$
• The integral of a constant function over an interval is the constant multiplied by the length of the interval.

#### 7. Integration by Parts for Definite Integrals:

• Statement: For two functions $u\left(x\right)$ and $v\left(x\right)$ differentiable on $\left[a,b\right]$, ${\int }_{a}^{b}u\left(x\right)\text{\hspace{0.17em}}dv\left(x\right)=\left[u\left(x\right)\cdot v\left(x\right)\right]{\mid }_{a}^{b}-{\int }_{a}^{b}v\left(x\right)\text{\hspace{0.17em}}du\left(x\right)$
• Integration by parts can be applied to definite integrals, but the boundary terms need to be accounted for in the evaluation.

#### 8. Change of Variable Property (Substitution Rule):

• Statement: If $u=g\left(x\right)$ is a differentiable function with a continuous derivative on $\left[a,b\right]$, ${\int }_{a}^{b}f\left(g\left(x\right)\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)\text{\hspace{0.17em}}dx={\int }_{g\left(a\right)}^{g\left(b\right)}f\left(u\right)\text{\hspace{0.17em}}du$
• If $u=g\left(x\right)$ is a differentiable function on $\left[a,b\right]$and $f\left(g\left(x\right)\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)$ is integrable on $\left[a,b\right]$, then ${\int }_{g\left(a\right)}^{g\left(b\right)}f\left(u\right)\text{\hspace{0.17em}}du={\int }_{a}^{b}f\left(g\left(x\right)\right)\cdot {g}^{\mathrm{\prime }}\left(x\right)\text{\hspace{0.17em}}dx$.
• This property helps in simplifying integrals by changing the variable of integration.

9. Inequality Property:

If $f\left(x\right)\le g\left(x\right)$ for all $x$ in $\left[a,b\right]$, then ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx\le {\int }_{a}^{b}g\left(x\right)\text{\hspace{0.17em}}dx$.

10. Absolute Value Property:

$\mathrm{\mid }{\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx\mathrm{\mid }\le {\int }_{a}^{b}\mathrm{\mid }f\left(x\right)\mathrm{\mid }\text{\hspace{0.17em}}dx$.

If a function $f\left(x\right)$ is integrable over subintervals $\left[a,b\right]$ and $\left[b,c\right]$, then ${\int }_{a}^{c}f\left(x\right)\text{\hspace{0.17em}}dx={\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx+{\int }_{b}^{c}f\left(x\right)\text{\hspace{0.17em}}dx$.
For a function $f\left(x\right)$ with period $T$, ${\int }_{a}^{a+T}f\left(x\right)\text{\hspace{0.17em}}dx={\int }_{0}^{T}f\left(x\right)\text{\hspace{0.17em}}dx$ for any $a$.