consider the function $ f(x) = 2x^9 + 10x^5 - 3x^3 - 8 $.

the antiderivative of $ f(x) $, where $ c = 0 $, is $ f(x) = ax^n + bx^m + cx^p + dx^q $, where

$ a $ is

and $ n $ is

and $ b $ is

and $ m $ is

and $ c $ is

and $ p $ is

and $ d $ is

and $ q $ is

Question

consider the function $ f(x) = 2x^9 + 10x^5 - 3x^3 - 8 $.

the antiderivative of $ f(x) $, where $ c = 0 $, is $ f(x) = ax^n + bx^m + cx^p + dx^q $, where

$ a $ is

and $ n $ is

and $ b $ is

and $ m $ is

and $ c $ is

and $ p $ is

and $ d $ is

and $ q $ is

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Antiderivatives", "Power Rule for Integration" ], "new_concepts": [], "current_concepts": [ "Antiderivatives", "Power Rule for Integration" ] } ## Integrate the polynomial function Using the *Antiderivatives* and *Power Rule for Integration* knowledge points $$ \begin{aligned} F(x) &= \int (2x^9 + 10x^5 - 3x^3 - 8) \, dx \ &= 2 \cdot \frac{x^{10}}{10} + 10 \cdot \frac{x^6}{6} - 3 \cdot \frac{x^4}{4} - 8x + C \end{aligned} $$ ## Simplify the coefficients Using the *Power Rule for Integration* knowledge point $$ \begin{aligned} F(x) &= \frac{1}{5}x^{10} + \frac{5}{3}x^6 - \frac{3}{4}x^4 - 8x + C \end{aligned} $$ ## Match with the given form Using the *Antiderivatives* knowledge point $$ \begin{aligned} &\text{Given } C = 0 \text{ and } F(x) = Ax^n + Bx^m + Cx^p + Dx^q:\ &A = \frac{1}{5}, \quad n = 10 \ &B = \frac{5}{3}, \quad m = 6 \ &C = -\frac{3}{4}, \quad p = 4 \ &D = -8, \quad q = 1 \end{aligned} $$ Consider the function $f(x) = 2x^9 + 10x^5 - 3x^3 - 8$. The antiderivative of $f(x)$, where $C = 0$, is $F(x) = Ax^n + Bx^m + Cx^p + Dx^q$, where $A$ is $\frac{1}{5}$ and $n$ is 10 and $B$ is $\frac{5}{3}$ and $m$ is 6 and $C$ is $-\frac{3}{4}$ and $p$ is 4 and $D$ is -8 and $q$ is 1 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Antiderivatives" ] }

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QUESTION IMAGE

Question

Explanation:

Integrate the polynomial function

Simplify the coefficients

Match with the given form

Answer: