Question

Need to evaluate integral pls help me

Answer 1

Explanation:

`\int\limits^1_0 {9x^9} \, dx +\int\limits^2_1 {4x^3} \, dx`

=
`(9x^(10))/(10) |_0^1 + x^4|_1^2`

`=(9)/(10)+2^4 -1 = 15(9)/(10)`

Answer 2

`\displaystyle \int\limits^2_0 {f(x)} \, dx = (159)/(10)`

General Formulas and Concepts:

Calculus

Integration

• Integrals
• Integral Notation

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + C`

Integration Rule [Fundamental Theorem of Calculus 1]:
`\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)`

Integration Property [Multiplied Constant]:
`\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx`

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

Integration Property [Splitting Integral]:
`\displaystyle \int\limits^c_a {f(x)} \, dx = \int\limits^b_a {f(x)} \, dx + \int\limits^c_b {f(x)} \, dx`

Explanation:

Step 1: Define

Identify.

`\displaystyle f(x) = \left \{ {{9x^9 ,\ 0 \leq x \leq 1} \atop {4x^3 ,\ 1 \leq x \leq 2}} \right.`

`\displaystyle \int\limits^2_0 {f(x)} \, dx = \ ?`

Step 2: Integrate

1. [Integral] Rewrite [Integration Property - Splitting Integral]:
   `\displaystyle \int\limits^2_0 {f(x)} \, dx = \int\limits^1_0 {f(x)} \, dx + \int\limits^2_1 {f(x)} \, dx`
2. [Integrand] Substitute in function:
   `\displaystyle \int\limits^2_0 {f(x)} \, dx = \int\limits^1_0 {9x^9} \, dx + \int\limits^2_1 {4x^3} \, dx`
3. [Integrals] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int\limits^2_0 {f(x)} \, dx = 9 \int\limits^1_0 {x^9} \, dx + 4 \int\limits^2_1 {x^3} \, dx`
4. [Integrals] Integration Rule [Reverse Power Rule]:
   `\displaystyle \int\limits^2_0 {f(x)} \, dx = 9 \bigg( (x^(10))/(10) \bigg) \bigg| \limits^1_0 + 4 \bigg( (x^4)/(4) \bigg) \bigg| \limits^2_1`
5. Integration Rule [Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^2_0 {f(x)} \, dx = 9 \bigg( (1)/(10) \bigg) + 4 \bigg( (15)/(4) \bigg)`
6. Simplify:
   `\displaystyle \int\limits^2_0 {f(x)} \, dx = (159)/(10)`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration