int x(x-1)^(5)dx

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int x(x-1)^(5)dx

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Recognize opportunity for integration by parts: Recognize the integral as an opportunity to use integration by parts. Integration by parts formula: ∫u dv = uv - ∫v du Let u = x and dv = (x-1)^5 dx.Differentiate u and integrate dv: Differentiate u and integrate dv. du = dx and v = ∫(x-1)^5 dx. To find v, we need to integrate (x-1)^5.Integrate (x-1)^5 using substitution: Integrate (x-1)^5 using the substitution method. Let t = x - 1, then dt = dx and when x = 1, t = 0. Now we integrate t^5 dt.Calculate integral of t^5: Calculate the integral of t^5. ∫t^5 dt = (1/6)t^6 + C Now we substitute back x - 1 for t. v = (1/6)(x - 1)^6 + CApply integration by parts formula: Apply the integration by parts formula. ∫x(x-1)^5 dx = uv - ∫v du = x * [(1/6)(x - 1)^6 + C] - ∫[(1/6)(x - 1)^6 + C] dxSimplify expression and integrate remaining term: Simplify the expression and integrate the remaining term. = (1/6)x(x - 1)^6 - (1/6)∫(x - 1)^6 dx Now we need to integrate (x - 1)^6.Integrate (x - 1)^6 using substitution: Integrate (x - 1)^6 using the substitution method. Let t = x - 1, then dt = dx and when x = 1, t = 0. Now we integrate t^6 dt.Calculate integral of t^6: Calculate the integral of t^6. ∫t^6 dt = (1/7)t^7 + C Now we substitute back x - 1 for t. ∫(x - 1)^6 dx = (1/7)(x - 1)^7 + CSubstitute integral back into expression: Substitute the integral back into the expression. = (1/6)x(x - 1)^6 - (1/6) * [(1/7)(x - 1)^7 + C] = (1/6)x(x - 1)^6 - (1/42)(x - 1)^7 - C/6Combine constants and write final answer: Combine the constants and write the final answer. The indefinite integral of x(x-1)^5 with respect to x is: I = (1/6)x(x - 1)^6 - (1/42)(x - 1)^7 + C Where C is the constant of integration.

$\int x(x-1)^{5} d x$

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∫x(x−1)5dx \int x(x-1)^{5} d x ∫x(x−1)5dx

$\int x(x-1)^{5} d x$