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Ber [7]
2 years ago
6

Need to evaluate integral pls help me

Mathematics
2 answers:
Lemur [1.5K]2 years ago
8 0

Answer:

Step-by-step explanation:

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

=\frac{9x^{10}}{10} |_0^1 + x^4|_1^2

=\frac{9}{10}+2^4 -1 = 15\frac{9}{10}

olga nikolaevna [1]2 years ago
4 0

Answer:

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

General Formulas and Concepts:

<u>Calculus</u>

Integration

  • Integrals
  • Integral Notation

Integration Rule [Reverse Power Rule]:                                                               \displaystyle \int {x^n} \, dx = \frac{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

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify.</em>

\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 = \ ?

<u>Step 2: Integrate</u>

  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( \frac{x^{10}}{10} \bigg) \bigg| \limits^1_0 + 4 \bigg( \frac{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( \frac{1}{10} \bigg) + 4 \bigg( \frac{15}{4} \bigg)
  6. Simplify:                                                                                                         \displaystyle \int\limits^2_0 {f(x)} \, dx = \frac{159}{10}

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

Unit: Integration

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