Question
Find the indefinite integral of \( f(x) = 6x^2 + 5x - 3 \) with respect to \( x \).
Ask by Coleman Page. in Turkey
Nov 21,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( 2x^{3}+\frac{5x^{2}}{2}-3x + C \)
Solution
Calculate the integral \( 6x^2 + 5x - 3 \).
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
\(\int 6x^{2}+5x-3 dx\)
- step1: Use properties of integrals:
\(\int 6x^{2} dx+\int 5x dx+\int -3 dx\)
- step2: Evaluate the integral:
\(2x^{3}+\int 5x dx+\int -3 dx\)
- step3: Evaluate the integral:
\(2x^{3}+\frac{5x^{2}}{2}+\int -3 dx\)
- step4: Evaluate the integral:
\(2x^{3}+\frac{5x^{2}}{2}-3x\)
- step5: Add the constant of integral C:
\(2x^{3}+\frac{5x^{2}}{2}-3x + C, C \in \mathbb{R}\)
The indefinite integral of \( f(x) = 6x^2 + 5x - 3 \) with respect to \( x \) is \( 2x^{3}+\frac{5x^{2}}{2}-3x + C \), where \( C \) is an arbitrary constant.
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Simplify this solution The Deep Dive
To find the indefinite integral of \( f(x) = 6x^2 + 5x - 3 \) with respect to \( x \), we integrate each term separately: \[ \int f(x) \, dx = \int (6x^2) \, dx + \int (5x) \, dx - \int (3) \, dx \] Calculating each integral: 1. \( \int (6x^2) \, dx = 6 \cdot \frac{x^3}{3} = 2x^3 \) 2. \( \int (5x) \, dx = 5 \cdot \frac{x^2}{2} = \frac{5}{2}x^2 \) 3. \( \int (-3) \, dx = -3x \) Combining these results gives: \[ \int f(x) \, dx = 2x^3 + \frac{5}{2}x^2 - 3x + C \] where \( C \) is the constant of integration. So, the indefinite integral is: \[ \int (6x^2 + 5x - 3) \, dx = 2x^3 + \frac{5}{2}x^2 - 3x + C \]
