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intx^(2)(6-x^(3))^(5)dx

∫x2(6−x3)5dx \int x^{2}\left(6-x^{3}\right)^{5} d x

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Full solution

Q. ∫x2(6−x3)5dx \int x^{2}\left(6-x^{3}\right)^{5} d x
  1. Rewrite in terms of uu: Rewrite the integral in terms of uu.∫x2(6−x3)5 dx=∫(6−u)5⋅(−13) du\int x^2(6-x^3)^5 \, dx = \int (6-u)^5 \cdot \left(-\frac{1}{3}\right) \, du
  2. Integrate with respect to uu: Now, integrate (6−u)5∗(−13)(6-u)^5 * (-\frac{1}{3}) with respect to uu.\newline∫(6−u)5∗(−13) du=(−13)∗∫(6−u)5 du\int (6-u)^5 * (-\frac{1}{3}) \, du = (-\frac{1}{3}) * \int (6-u)^5 \, du
  3. Expand and integrate term by term: Expand (6−u)5(6-u)^5 and integrate term by term.(−13)×∫(6−u)5 du=(−13)×[656×u−5×647×u2+10×638×u3−10×629×u4+5×610×u5−u66]+C\left(-\frac{1}{3}\right) \times \int (6-u)^5 \, du = \left(-\frac{1}{3}\right) \times \left[\frac{6^5}{6} \times u - \frac{5\times6^4}{7} \times u^2 + \frac{10\times6^3}{8} \times u^3 - \frac{10\times6^2}{9} \times u^4 + \frac{5\times6}{10} \times u^5 - \frac{u^6}{6}\right] + C

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