Mathematics, 20.09.2020 14:01 Katmcfee7681

Evaluate the definite integral.
1 x4(1 + 2x5)5 dx.

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Answer from: tilly40oooo

$\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{9768748}{5}$

General Formulas and Concepts:

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Multiplied Constant]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]: $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integrals

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]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx$

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

Set u: $\displaystyle u = 1 + 2x^5$ [u] Differentiate [Basic Power Rule, Derivative Properties]: $\displaystyle du = 10x^4 \ dx$ [Bounds] Switch: $\displaystyle \left \{ {{x = 5,\ u = 1 + 2(5)^5 = 6251} \atop {x=1,\ u = 1 + 2(1)^5 = 3}} \right.$

Step 3: Integrate Pt. 2

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{1}{10}\int\limits^5_1 {10x^4(1 + 2x^5)} \, dx$ [Integral] U-Substitution: $\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{1}{10}\int\limits^{6251}_3 {u} \, du$ [Integral] Reverse Power Rule: $\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{1}{10} \bigg( \frac{u^2}{2} \bigg) \bigg| \limits^{6251}_3$ Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{1}{10}(19537496)$ Simplify: $\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{9768748}{5}$

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

Unit: Integration

Answer from: Quest

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step-by-step explanation:

Answer from: Quest

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Evaluate the definite integral.
1 x4(1 + 2x5)5 dx....