Example Question - integral of polynomial function
Here are examples of questions we've helped users solve.
Integral of Polynomial Function
The integral in the image is an indefinite integral of a polynomial function. To solve the integral, you would integrate each term separately, applying the power rule for integration. Here's the integration term-by-term: ∫(8x^3 - x^2 + 5x - 1) dx = ∫8x^3 dx - ∫x^2 dx + ∫5x dx - ∫1 dx Now applying the power rule for integration (which states that ∫x^n dx = x^(n+1)/(n+1) for all n ≠ -1), we get: = 8 * x^(3+1)/(3+1) - x^(2+1)/(2+1) + 5 * x^(1+1)/(1+1) - x + C = 8 * x^4/4 - x^3/3 + 5 * x^2/2 - x + C Simplifying, we have: = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C So the integral is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C where C represents the constant of integration.
Integral of a Polynomial Function
The integral in the image is: ∫ (8x^3 - x^2 + 5x - 1) dx To solve the integral, integrate each term separately with respect to x: ∫ 8x^3 dx = (8/4)x^4 = 2x^4 ∫ -x^2 dx = -(1/3)x^3 = -x^3/3 ∫ 5x dx = (5/2)x^2 = 5x^2/2 ∫ -1 dx = -x Now, combine the integrated terms: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Here, C is the constant of integration. The final answer for the integral is: 2x^4 - (x^3/3) + (5x^2/2) - x + C
Antiderivative of a Polynomial Function
Certainly! The expression you provided is an integral, and we will find the antiderivative (indefinite integral) of the function. Here's the integral: \[ \int (8x^3 - x^2 + 5x - 1) \, dx \] To solve this integral, we will integrate each term separately. Recall the power rule for integration, \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) where C is the constant of integration. Applying this rule to each term: 1. The integral of \(8x^3\) is \(\frac{8}{4}x^{4} = 2x^4\). 2. The integral of \(-x^2\) is \(-\frac{1}{3}x^{3} = -\frac{1}{3}x^3\). 3. The integral of \(5x\) is \(\frac{5}{2}x^{2} = \frac{5}{2}x^2\). 4. The integral of \(-1\) is \(-x\). Now, putting it all together and adding the constant of integration, \(C\), the antiderivative is: \[ \int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
Solving Integral of a Polynomial Function
The image shows an integral that you'd like to solve: \[ \int (8x^3 - x^2 + 5x - 1) \, dx \] To solve this integral, integrate each term individually with respect to \( x \). When integrating a polynomial term of the form \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} \) provided that \( n \neq -1 \). \[ \int 8x^3 \, dx = \frac{8}{4}x^{4} = 2x^4 \] \[ \int (-x^2) \, dx = -\frac{1}{3}x^{3} \] \[ \int 5x \, dx = \frac{5}{2}x^{2} \] \[ \int (-1) \, dx = -x \] Now summing all of the individual antiderivatives and adding a constant of integration \( C \), we have: \[ 2x^4 - \frac{1}{3}x^{3} + \frac{5}{2}x^{2} - x + C \] So the integral of \( 8x^3 - x^2 + 5x - 1 \) with respect to \( x \) is: \[ \int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^{3} + \frac{5}{2}x^{2} - x + C \]
Integration of a Polynomial Function
To solve the given integral, we will integrate each term separately: \[ \int (8x^3 - x^2 + 5x - 1)\ dx \] We integrate each term using the power rule for integration, which is: \[ \int x^n\ dx = \frac{x^{n+1}}{n+1} + C \] where \(n \neq -1\) and \(C\) is the constant of integration. Now, integrating each term: 1. \(\int 8x^3\ dx = \frac{8}{4}x^{3+1} = 2x^4\) 2. \(\int -x^2\ dx = -\frac{1}{3}x^{2+1} = -\frac{1}{3}x^3\) 3. \(\int 5x\ dx = \frac{5}{2}x^{1+1} = \frac{5}{2}x^2\) 4. \(\int -1\ dx = -x\) Combine these results and add the constant of integration (\(C\)): \[ \int (8x^3 - x^2 + 5x - 1)\ dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] So, the integral of \(8x^3 - x^2 + 5x - 1\) with respect to \(x\) is: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com