int(2x+6)^(5)dx

Question

int(2x+6)^(5)dx

Bytelearn · Accepted Answer

Recognize the integral: Recognize the integral to be solved. We need to find the integral of (2x+6)^5 with respect to x.Apply power rule: Apply the power rule for integration. The power rule for integration states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, where C is the constant of integration. However, since we have a linear term (2x+6) instead of just x, we need to adjust the rule accordingly.Use substitution: Use substitution to simplify the integral. Let u = 2x + 6, then du = 2dx. This means that dx = du/2.Rewrite in terms of u: Rewrite the integral in terms of u. The integral becomes (1/2) * ∫u^5 du, since we have to divide by 2 to account for the substitution of dx.Apply power rule for u: Apply the power rule for integration to the integral in terms of u. Using the power rule, we get (1/2) * (u^6/6) + C.Substitute back x: Substitute back the original variable x into the integral. Replace u with 2x + 6 to get (1/2) * ((2x + 6)^6/6) + C.Simplify expression: Simplify the expression. We can simplify the expression to (1/12) * (2x + 6)^6 + C.

$\int(2x+6)^5\,dx$

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∫(2x+6)5 dx\int(2x+6)^5\,dx∫(2x+6)5dx

$\int(2x+6)^5\,dx$