Example Question - power rule for integration
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.
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: \[ \int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
Integration of Polynomial Functions
To solve the given integral, we will integrate each term separately. The integral given is: ∫(8x^3 - x^2 + 5x - 1)dx To integrate this, we use the power rule for integration: ∫x^n dx = x^(n+1) / (n+1) + C, where n ≠ -1 and C is the constant of integration. Applying this rule for each term, we get: ∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1) / (3+1)) + C_1 = 8 * (x^4 / 4) + C_1 = 2x^4 + C_1 ∫(-x^2) dx = - ∫x^2 dx = - (x^(2+1) / (2+1)) + C_2 = - (x^3 / 3) + C_2 ∫5x dx = 5 * ∫x dx = 5 * (x^(1+1) / (1+1)) + C_3 = 5 * (x^2 / 2) + C_3 = (5/2)x^2 + C_3 ∫(-1) dx = -x + C_4 Now, adding them all together and combining the constants of integration, we get: ∫(8x^3 - x^2 + 5x - 1)dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Where C is the constant of integration, representing that there can be any constant added to the integral (C = C_1 + C_2 + C_3 + C_4).
Integration of a Polynomial
Certainly! You are asked to integrate the given polynomial. Here’s how you do it: The integral of a polynomial is found by integrating each term individually. The integral of \( x^n \) with respect to \( x \) is \( \frac{x^{n+1}}{n+1} \) provided \( n \neq -1 \). Let’s integrate each term of the given polynomial: \[ \int (8x^3 - x^2 + 5x - 1) dx \] Integrating term by term: \[ = 8 \int x^3 dx - \int x^2 dx + 5 \int x dx - \int 1 dx \] Applying the power rule for integration: \[ = 8 \cdot \frac{x^{3+1}}{3+1} - \frac{x^{2+1}}{2+1} + 5 \cdot \frac{x^{1+1}}{1+1} - x + C \] Simplifying the expressions: \[ = 8 \cdot \frac{x^4}{4} - \frac{x^3}{3} + 5 \cdot \frac{x^2}{2} - x + C \] Further simplifying, which also involves reducing the fraction \(8 \cdot \frac{x^4}{4}\) to \(2x^4\): \[ = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] So the integral of \( 8x^3 - x^2 + 5x - 1 \) with respect to \( x \) is \( 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \), where \( C \) is the constant of integration.
Integration of Polynomial Function
Certainly! You need to integrate the given polynomial function with respect to x. The integration of a polynomial function is done term by term. Here are the steps: Given function: ∫(8x^3 - x^2 + 5x - 1)dx 1. Integrate each term separately using the power rule for integration: ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. For the first term 8x^3: ∫8x^3 dx = 8 * (x^(3+1))/(3+1) = 8 * x^4/4 = 2x^4 For the second term -x^2: ∫(-x^2) dx = - (x^(2+1))/(2+1) = -x^3/3 For the third term 5x: ∫5x dx = 5 * (x^(1+1))/(1+1) = 5/2 x^2 For the fourth term -1: ∫(-1) dx = -x 2. Combine the integrated terms and include the constant of integration: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Therefore, the indefinite integral of the given function 8x^3 - x^2 + 5x - 1 with respect to x is: 2x^4 - (1/3)x^3 + (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: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
Integrating a Polynomial using Power Rule
You are asked to integrate a polynomial. The integral of a polynomial is found by integrating each term separately, according to the power rule for integration, \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) where \(C\) is the constant of integration. The integral of the polynomial \(8x^3 - x^2 + 5x - 1\) is calculated as follows: \[ \int (8x^3 - x^2 + 5x - 1)dx = \int 8x^3 dx - \int x^2 dx + \int 5x dx - \int 1dx \] Now integrate each term separately: \[ \int 8x^3 dx = 8 \times \frac{x^{3+1}}{3+1} = 2x^4 \] \[ \int x^2 dx = \frac{x^{2+1}}{2+1} = \frac{1}{3}x^3 \] \[ \int 5x dx = 5 \times \frac{x^{1+1}}{1+1} = \frac{5}{2}x^2 \] \[ \int 1dx = x \] So the integral is: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] where \(C\) is the constant of integration.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com