Answer:
[tex]\displaystyle \int {4x^6 - 2x^3 + 7x - 4} \, dx = \boxed{ \frac{4x^7}{7} - \frac{x^4}{2} + \frac{7x^2}{2} - 4x + C }[/tex]
General Formulas and Concepts:
Calculus
Integration
Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Step-by-step explanation:
Step 1: Define
Identify given.
[tex]\displaystyle \int {4x^6 - 2x^3 + 7x - 4} \, dx[/tex]
Step 2: Integrate
- [Integral] Rewrite [Integration Property - Addition/Subtraction]:
[tex]\displaystyle \int {4x^6 - 2x^3 + 7x - 4} \, dx = \int {4x^6} \, dx - \int {2x^3} \, dx + \int {7x} \, dx - \int {4} \, dx[/tex] - [Integrals] Rewrite [Integration Property - Multiplied Constant]:
[tex]\displaystyle \int {4x^6 - 2x^3 + 7x - 4} \, dx = 4 \int {x^6} \, dx - 2 \int {x^3} \, dx + 7 \int {x} \, dx - 4 \int {} \, dx[/tex] - [Integrals] Apply Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {4x^6 - 2x^3 + 7x - 4} \, dx = 4 \bigg( \frac{x^7}{7} \bigg) - 2 \bigg( \frac{x^4}{4} \bigg) + 7 \bigg( \frac{x^2}{2} \bigg) - 4x + C[/tex] - Simplify:
[tex]\displaystyle \int {4x^6 - 2x^3 + 7x - 4} \, dx = \boxed{ \frac{4x^7}{7} - \frac{x^4}{2} + \frac{7x^2}{2} - 4x + C }[/tex]
∴ we have found the given integral.
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
