解
∫6x5(x3+2)15dx
解
6(51x51+85x48+328x45+3260x42+560x39+38008x36+329120x33+27456x30+3183040x27+3320320x24+146432x21+3465920x18+3372736x15+71680x12+381920x9+316384x6)+C
解答ステップ
∫6x5(x3+2)15dx
定数を除く: ∫a⋅f(x)dx=a⋅∫f(x)dx=6⋅∫x5(x3+2)15dx
拡張 x5(x3+2)15:x50+30x47+420x44+3640x41+21840x38+96096x35+320320x32+823680x29+1647360x26+2562560x23+3075072x20+2795520x17+1863680x14+860160x11+245760x8+32768x5
=6⋅∫x50+30x47+420x44+3640x41+21840x38+96096x35+320320x32+823680x29+1647360x26+2562560x23+3075072x20+2795520x17+1863680x14+860160x11+245760x8+32768x5dx
総和規則を適用する: ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx=6(∫x50dx+∫30x47dx+∫420x44dx+∫3640x41dx+∫21840x38dx+∫96096x35dx+∫320320x32dx+∫823680x29dx+∫1647360x26dx+∫2562560x23dx+∫3075072x20dx+∫2795520x17dx+∫1863680x14dx+∫860160x11dx+∫245760x8dx+∫32768x5dx)
∫x50dx=51x51
∫30x47dx=85x48
∫420x44dx=328x45
∫3640x41dx=3260x42
∫21840x38dx=560x39
∫96096x35dx=38008x36
∫320320x32dx=329120x33
∫823680x29dx=27456x30
∫1647360x26dx=3183040x27
∫2562560x23dx=3320320x24
∫3075072x20dx=146432x21
∫2795520x17dx=3465920x18
∫1863680x14dx=3372736x15
∫860160x11dx=71680x12
∫245760x8dx=381920x9
∫32768x5dx=316384x6
=6(51x51+85x48+328x45+3260x42+560x39+38008x36+329120x33+27456x30+3183040x27+3320320x24+146432x21+3465920x18+3372736x15+71680x12+381920x9+316384x6)
定数を解答に追加する=6(51x51+85x48+328x45+3260x42+560x39+38008x36+329120x33+27456x30+3183040x27+3320320x24+146432x21+3465920x18+3372736x15+71680x12+381920x9+316384x6)+C