Topic

# Properties of Definite Integrals

Topic Progress:

\begin{aligned} \displaystyle \int_{a}^{b}{f(x)}dx &= -\int_{b}^{a}{f(x)}dx \\ \int_{a}^{a}{f(x)}dx &= 0 \\ \int_{-a}^{a}{\text{(Even Function)}}dx &= 2\int_{0}^{a}{\text{(Even Function)}}dx \\ \int_{-a}^{a}{\text{(Odd Function)}}dx &= 0 \\ \int_{a}^{c}{f(x)}dx + \int_{c}^{b}{f(x)}dx &= \int_{a}^{b}{f(x)}dx \\ \end{aligned}

## Practice Questions

### Question 1

Find $\displaystyle \int_{3}^{3}{(x^3 + 6x^2)}dx$.
\begin{aligned} \displaystyle \require{color} \int_{3}^{3}{(x^3 + 6x^2)}dx &= 0 &\color{red} \because \int_{a}^{a}{f(x)}dx = 0 \\ \end{aligned}

### Question 2

Find $k$, if $\displaystyle \int_{9}^{k}{(x^5 + 6x^3)}dx = 0$.
\begin{aligned} \displaystyle \require{color} \int_{9}^{\color{red}9}{(x^5 + 6x^3)}dx &= 0 \\ \therefore k &= 9 &\color{red} \because \int_{a}^{a}{f(x)}dx = 0 \end{aligned}

### Question 3

Find $k$, if $\displaystyle \int_{1}^{2}{4x^3}dx + \int_{2}^{4}{4x^3}dx = \int_{1}^{k}{4x^3}dx$.
\begin{aligned} \displaystyle \require{color} \int_{1}^{2}{4x^3}dx + \int_{2}^{4}{4x^3}dx &= \int_{1}^{\color{red}4}{4x^3}dx \ \ \ \ \color{red} \because \int_{a}^{c}{f(x)}dx + \int_{c}^{b}{f(x)}dx = \int_{a}^{b}{f(x)}dx \\ \therefore k &= 4 \end{aligned}

### Question 4

Find $k$, if $\displaystyle \int_{3}^{5}{(5x^5+4x^4)}dx = -\int_{k}^{3}{(5x^5+4x^4)}dx$.
\begin{aligned} \displaystyle \require{color} \int_{3}^{5}{(5x^5+4x^4)}dx &= -\int_{\color{red}5}^{3}{(5x^5+4x^4)}dx \color{red} \ \ \ \color{red} \because \int_{a}^{b}{f(x)}dx = -\int_{b}^{a}{f(x)}dx \\ \therefore k &= 5 \end{aligned}

### Question 5

Find $\displaystyle \int_{-1}^{1}{(7x^6+3x^2)}dx$ by identifying either even or odd function.
\begin{aligned} \displaystyle \require{color} f(x) &= 7x^6+ 3x^2 \\ f(-x) &= 7(-x)^6+ 3(-x)^2 \\ &= 7x^6+ 3x^2 \\ f(-x) &= f(x) \\ f(x) &\text{ is even}. \\ \int_{-1}^{1}{(7x^6+3x^2)}dx &= 2\int_{0}^{1}{(7x^6+3x^2)}dx \ \ \ \ \color{red} \int_{-a}^{a}{\text{(Even Function)}}dx = 2\int_{0}^{a}{\text{(Even Function)}}dx \\ &= 2\bigg[\dfrac{7x^{6+1}}{6+1} + \dfrac{3x^{2+1}}{2+1}\bigg]_{0}^{1} \\ &= 2\big[x^7 + x^3\big]_{0}^{1} \\ &= 2\big[(1^7 + 1^3)-(0^7 + 0^3)\big] \\ &= 4 \end{aligned}

### Question 6

Find $\displaystyle \int_{-4}^{4}{(6x^5+2x)}dx$ by identifying either even or odd function.
\begin{aligned} \displaystyle \require{color} f(x) &= 6x^5+2x \\ f(-x) &= 6(-x)^5+2(-x) \\ &= -6x^5 - 2x \\ &= -f(x) \\ f(-x) &= -f(x) \\ f(x) &\text{ is odd}. \ \ \ \ \color{red} \int_{-a}^{a}{\text{(Odd Function)}}dx = 0 \\ \therefore \int_{-4}^{4}{(6x^5+2x)}dx &= 0 \end{aligned}