integrate (20y ^ 3 + 10y) dx from 0 to 2
Since the integration is with respect to x, we can treat y as a constant. Thus, we can integrate 20y^3 with respect to x to get 20y^3x, and integrate 10y with respect to x to get 10yx. Then, we can evaluate the expression at x=2 and x=0, and subtract the two values to get the final answer.
∫(20y^3 + 10y) dx from 0 to 2
= [20y^3x + 10yx] from 0 to 2
= [20y^3(2) + 10y(2)] - [20y^3(0) + 10y(0)]
= 40y^3 + 20y - 0 - 0
= 40y^3 + 20y
Therefore, the value of the integral is 40y^3 + 20y.
Integrate 20y³ dx
Integrating 20y³ with respect to x means treating y as a constant and integrating with respect to x, so the integral of 20y³ dx is:
20y³x + C
where C is the constant of integration.
To integrate the expression (20y^3 + 10y) with respect to x from 0 to 2, there seems to be a discrepancy since the expression is given in terms of y, but the integral is with respect to x. Could you please clarify whether you meant to integrate this expression with respect to y or if there was a typo?
To integrate the expression (20y^3 + 10y) dx from 0 to 2, you can follow these steps:
Step 1: Recognize that y is a constant in this case since we are taking the integral with respect to x. So we can treat it as a constant and integrate with respect to x.
Step 2: Apply the power rule for integration. For a term of the form x^n, the integral is (1/(n+1)) * x^(n+1). In this case, we have two terms: 20y^3 and 10y.
Integrating the first term (20y^3), we get:
(1/4) * 20y^3 * x^4 + C1
Integrating the second term (10y), we get:
10y * x + C2
Step 3: Combine the integrals from Step 2 and apply the limits of integration. Since we are integrating with respect to x, the limits of integration are 0 and 2 in this case.
Applying the limits to the first term:
[(1/4) * 20y^3 * 2^4 - (1/4) * 20y^3 * 0^4] + C1
Simplifying further:
(1/4) * 20y^3 * 16 - (1/4) * 20y^3 * 0 + C1
Applying the limits to the second term:
[10y * 2 - 10y * 0] + C2
Simplifying further:
20y - 0 + C2
Step 4: Combine the results from the first and second terms to obtain the final result:
[(1/4) * 20y^3 * 16 - (1/4) * 20y^3 * 0] + [10y * 2 - 10y * 0]
Simplifying further:
(1/4) * 20y^3 * 16 + 20y
The final result after applying the limits of integration and simplifying is:
(1/4) * 20y^3 * 16 + 20y
Please note that C1 and C2 represent constants of integration, which could be determined if additional information about the problem is given.