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Compute an integral of x^2 (x-5)^4

by RoRi

Compute

    \[ \int_0^5 x^2 (x-5)^4 \, dx. \]

First, we’ll simplify our lives (slightly) by applying the fact that integrals are invariant under translation (Apostol, Theorem 1.18 with c=5) to get

    \[ \int_0^5 x^2 (x-5)^4 \, dx = \int_{-5}^0 (x+5)^2 x^4 \, dx. \]

Then, we expand and evaluate (our translation bought us having to expand (x+5)^2 instead of (x-5)^4: a small improvement).

    \begin{align*} \int_{-5}^0 (x+5)^2 x^4 \, dx &= \int_{-5}^0 (x^6 + 10x^5 +25x^4) \, dx \\ &= \left. \left( \frac{x^7}{7} + 10 \frac{x^6}{6} + 25 \frac{x^5}{5} \right) \right|_{-5}^0 \\ &= 0 - \left( -\frac{5^7}{7} + \frac{5^7}{3} - 5^6 \right) \\ &= \frac{3 \cdot 5^7 - 7 \cdot 5^7 + 21 \cdot 5^6}{21} \\ &= \frac{-20\cdot 5^6 + 21 \cdot 5^6}{21} \\ &= \frac{5^6}{21}. \end{align*}

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