Example Question - integration of polynomial function
Here are examples of questions we've helped users solve.
Integration of Polynomial Function
To solve the integral provided in the image, we will integrate each term separately. The integral in question is: ∫(8x^3 - x^2 + 5x - 1)dx Each term in the integral can be integrated by increasing the exponent by one and dividing by the new exponent. Don't forget to add the constant of integration at the end, which is typically denoted as 'C'. Here's how it is done step by step: ∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1) / (3+1)) = 8 * (x^4 / 4) = 2x^4 ∫(-x^2) dx = -1 * ∫x^2 dx = -1 * (x^(2+1) / (2+1)) = -1 * (x^3 / 3) = -x^3 / 3 ∫5x dx = 5 * ∫x dx = 5 * (x^(1+1) / (1+1)) = 5 * (x^2 / 2) = (5/2) * x^2 ∫(-1) dx = -x Now, summing up all parts and adding the constant of integration, C, we get: 2x^4 - (x^3 / 3) + (5/2) * x^2 - x + C So, the antiderivative of the function 8x^3 - x^2 + 5x - 1 is: 2x^4 - (x^3 / 3) + (5/2) * x^2 - x + C
Integration of a Polynomial Function
Sure, the given integral is: ∫ (8x^3 - x^2 + 5x - 1) dx We need to integrate each term separately. The integral of a polynomial function is found by increasing the exponent by one and dividing by the new exponent. Here's how it works for each term: 1. Integral of 8x^3 dx: Increase exponent by 1 (from 3 to 4), then divide by the new exponent (4). ∫ 8x^3 dx = 8/4 x^4 = 2x^4 2. Integral of -x^2 dx: Increase exponent by 1 (from 2 to 3), then divide by the new exponent (3). ∫ -x^2 dx = -1/3 x^3 3. Integral of 5x dx: Increase exponent by 1 (from 1 to 2), then divide by the new exponent (2). ∫ 5x dx = 5/2 x^2 4. Integral of -1 dx: Since the exponent is 0 (because -1 is the same as -1x^0), we just multiply x by the constant. ∫ -1 dx = -1 * x = -x Now, let's put it all together: ∫ (8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Don't forget to add the constant of integration, C, at the end. The final answer is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com