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Evaluate the integral. int p(p+4)^(5)dp

Evaluate the integral.\newline∫p(p+4)5dp \int p(p+4)^{5} d p

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Find indefinite integrals using the substitution and by partsright chevron icon

Full solution

Q. Evaluate the integral.\newline∫p(p+4)5dp \int p(p+4)^{5} d p
  1. Substitution: Let's do a substitution. Let u=p+4u = p + 4, then du=dpdu = dp.
  2. Change in terms of u: Now we need to change pp in terms of uu. Since u=p+4u = p + 4, p=u−4p = u - 4.
  3. Substitute and expand: Substitute pp and dpdp into the integral: ∫(u−4)u5 du\int (u - 4)u^5 \, du.
  4. Integrate terms: Expand the integrand: ∫(u6−4u5) du\int (u^6 - 4u^5) \, du.
  5. Simplify integral: Integrate term by term: (u77)−(4u66)+C(\frac{u^7}{7}) - (\frac{4u^6}{6}) + C.
  6. Substitute back u: Simplify the integral: u77\frac{u^7}{7} - 2u63\frac{2u^6}{3} + CC.
  7. Substitute back uu: Simplify the integral: u77\frac{u^7}{7} - 2u63\frac{2u^6}{3} + CC.Substitute back u=p+4u = p + 4: (p+4)77\frac{(p + 4)^7}{7} - 2(p+4)63\frac{2(p + 4)^6}{3} + CC.

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