以下の定積分を計算する問題です。 $\int_{1}^{3} x^2(x-4) dx + 4\int_{1}^{3} x(x-1) dx - \int_{2}^{3} x(x+2)(x-2) dx$解析学定積分積分計算多項式2025/7/51. 問題の内容以下の定積分を計算する問題です。∫13x2(x−4)dx+4∫13x(x−1)dx−∫23x(x+2)(x−2)dx\int_{1}^{3} x^2(x-4) dx + 4\int_{1}^{3} x(x-1) dx - \int_{2}^{3} x(x+2)(x-2) dx∫13x2(x−4)dx+4∫13x(x−1)dx−∫23x(x+2)(x−2)dx2. 解き方の手順まず、各積分を展開して整理します。∫13(x3−4x2)dx+4∫13(x2−x)dx−∫23x(x2−4)dx\int_{1}^{3} (x^3 - 4x^2) dx + 4\int_{1}^{3} (x^2 - x) dx - \int_{2}^{3} x(x^2 - 4) dx∫13(x3−4x2)dx+4∫13(x2−x)dx−∫23x(x2−4)dx∫13(x3−4x2)dx+4∫13(x2−x)dx−∫23(x3−4x)dx\int_{1}^{3} (x^3 - 4x^2) dx + 4\int_{1}^{3} (x^2 - x) dx - \int_{2}^{3} (x^3 - 4x) dx∫13(x3−4x2)dx+4∫13(x2−x)dx−∫23(x3−4x)dx次に、それぞれの積分を計算します。∫13(x3−4x2)dx=[14x4−43x3]13=(814−1083)−(14−43)=804−1043=20−1043=60−1043=−443\int_{1}^{3} (x^3 - 4x^2) dx = [\frac{1}{4}x^4 - \frac{4}{3}x^3]_{1}^{3} = (\frac{81}{4} - \frac{108}{3}) - (\frac{1}{4} - \frac{4}{3}) = \frac{80}{4} - \frac{104}{3} = 20 - \frac{104}{3} = \frac{60-104}{3} = -\frac{44}{3}∫13(x3−4x2)dx=[41x4−34x3]13=(481−3108)−(41−34)=480−3104=20−3104=360−104=−3444∫13(x2−x)dx=4[13x3−12x2]13=4[(273−92)−(13−12)]=4[(9−92)−(2−36)]=4[92+16]=4[27+16]=4[286]=4[143]=5634\int_{1}^{3} (x^2 - x) dx = 4[\frac{1}{3}x^3 - \frac{1}{2}x^2]_{1}^{3} = 4[(\frac{27}{3} - \frac{9}{2}) - (\frac{1}{3} - \frac{1}{2})] = 4[(9 - \frac{9}{2}) - (\frac{2-3}{6})] = 4[\frac{9}{2} + \frac{1}{6}] = 4[\frac{27+1}{6}] = 4[\frac{28}{6}] = 4[\frac{14}{3}] = \frac{56}{3}4∫13(x2−x)dx=4[31x3−21x2]13=4[(327−29)−(31−21)]=4[(9−29)−(62−3)]=4[29+61]=4[627+1]=4[628]=4[314]=356∫23(x3−4x)dx=[14x4−2x2]23=(814−18)−(4−8)=(814−18)+4=814−14=81−564=254\int_{2}^{3} (x^3 - 4x) dx = [\frac{1}{4}x^4 - 2x^2]_{2}^{3} = (\frac{81}{4} - 18) - (4 - 8) = (\frac{81}{4} - 18) + 4 = \frac{81}{4} - 14 = \frac{81 - 56}{4} = \frac{25}{4}∫23(x3−4x)dx=[41x4−2x2]23=(481−18)−(4−8)=(481−18)+4=481−14=481−56=425したがって、−443+563−254=123−254=4−254=16−254=−94-\frac{44}{3} + \frac{56}{3} - \frac{25}{4} = \frac{12}{3} - \frac{25}{4} = 4 - \frac{25}{4} = \frac{16 - 25}{4} = -\frac{9}{4}−344+356−425=312−425=4−425=416−25=−493. 最終的な答え−94-\frac{9}{4}−49