Integral
Okay, so I need to figure out how to solve this integral. Let me see... The problem is written as ∫(3x² + 2x - 5) dx. Alright, first, I remember that integrating a polynomial term by term is the way to go. Each term can be integrated separately, right? Let me recall the power rule for integration. The power rule says that the integral of x^n dx is (x^(n+1))/(n+1) + C, where C is the constant of integration.
So, let's break down the integral into three separate integrals. That would be ∫3x² dx + ∫2x dx - ∫5 dx. Now, for each of these, I can apply the power rule.
Starting with the first term: ∫3x² dx. The coefficient 3 can be factored out of the integral, so it's 3 times ∫x² dx. Applying the power rule here, the integral of x² is (x³)/3. So multiplying by the 3 in front, that would be 3*(x³/3) which simplifies to x³. Okay, that seems straightforward.
Next term: ∫2x dx. Similarly, factor out the 2, so it's 2∫x dx. The integral of x is (x²)/2. Multiplying by 2 gives 2(x²/2) which simplifies to x². Nice, that cancels out the coefficient again.
Third term: -∫5 dx. The integral of a constant is just the constant times x, right? So ∫5 dx is 5x, and with the negative sign, it becomes -5x.
Putting all those together: x³ + x² - 5x + C. Wait, don't forget the constant of integration! Since the original integral is indefinite, we need to add a constant C at the end.
Let me double-check each term. First term: derivative of x³ is 3x², which matches the original integrand's first term. Second term: derivative of x² is 2x, which matches the second term. Third term: derivative of -5x is -5, which matches the third term. And the constant C differentiates to zero, which is correct.
Hmm, seems like everything checks out. But wait, just to make sure I didn't make any mistakes with the coefficients. Let me go through each step again.
For ∫3x² dx: 3 stays outside, integrate x² to x³/3, multiply by 3 gives x³. Correct.
For ∫2x dx: 2 stays outside, integrate x to x²/2, multiply by 2 gives x². Correct.
For ∫5 dx: integral is 5x, so with the negative sign it's -5x. Correct.
Yep, all steps look good. So combining them all, the integral is x³ + x² - 5x + C. I think that's the final answer.
Wait, maybe I should also consider if there's any substitution or method needed here, but since it's a simple polynomial, term-by-term integration using the power rule suffices. There's no need for substitution or integration by parts. So I think I'm confident with this result.
Another way to verify is to differentiate the answer and see if we get back the original function. Let's differentiate x³ + x² - 5x + C. The derivative of x³ is 3x², derivative of x² is 2x, derivative of -5x is -5, and derivative of C is 0. So altogether, 3x² + 2x -5, which matches the original integrand. Perfect! That confirms that the integral is correct.
So, no mistakes detected. The integral of 3x² + 2x -5 with respect to x is x³ + x² -5x + C. I think that's all there is to it. Hopefully, I didn't overlook anything. Sometimes with integrals, especially definite integrals, you have to be careful about limits, but since this is indefinite, just adding the constant is enough. Alright, I think I'm done here.
The integral of the polynomial ( 3x^2 + 2x - 5 ) can be solved by integrating each term separately using the power rule. Here's the step-by-step solution:
Integrate each term individually:
For ( \int 3x^2 , dx ):
[
3 \cdot \frac{x^{3}}{3} = x^3
]
For ( \int 2x , dx ):
[
https://www.18122.com/url/usdt/usdt1813.jpg
2 \cdot \frac{x^{2}}{2} = x^2
]
For ( \int -5 , dx ):
[
-5x
]
Combine the results and add the constant of integration ( C ):
[
\int (3x^2 + 2x - 5) , dx = x^3 + x^2 - 5x + C
]
Verification: Differentiating ( x^3 + x^2 - 5x + C ) yields ( 3x^2 + 2x - 5 ), confirming the solution is correct.
Final Answer:
[
\boxed{x^3 + x^2 - 5x + C}
]
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