降次积分法

降次积分法是求高次函数积分的一种技巧。先用换元积分法、三角换元法、分部积分法、部分分式積分法等方法求出降次公式，将原函数（如In）用低次的函数形式（如In-2）表示。然后将n代成想求的数，逐步降次，直至降至0或1为止，借助积分表得出结果。 如在求 ∫ cos 5 ⁡ ( x ) d x {\displaystyle \int \cos ^{5}(x)\,dx\!} 时，需要先求得 ∫ cos n ⁡ ( x ) d x {\displaystyle \int \cos ^{n}(x)\,dx\!} 的降次公式，过程如下： I n = ∫ cos n ⁡ ( x ) d x {\displaystyle I_{n}\,=\int \cos ^{n}(x)\,dx\!} = ∫ cos n − 1 ⁡ ( x ) cos ⁡ ( x ) d x {\displaystyle =\int \cos ^{n-1}(x)\cos(x)\,dx\!} = ∫ cos n − 1 ⁡ ( x ) d ( sin ⁡ ( x ) ) {\displaystyle =\int \cos ^{n-1}(x)\,d(\sin(x))\!} = cos n − 1 ⁡ ( x ) sin ⁡ ( x ) − ∫ sin ⁡ ( x ) d ( c o s n − 1 ( x ) ) {\displaystyle =\cos ^{n-1}(x)\sin(x)-\int \sin(x)\,d(cos^{n-1}(x))\!} = cos n − 1 ⁡ ( x ) sin ⁡ ( x ) + ( n − 1 ) ∫ sin ⁡ ( x ) cos n − 2 ⁡ ( x ) sin ⁡ ( x ) d x {\displaystyle =\cos ^{n-1}(x)\sin(x)+(n-1)\int \sin(x)\cos ^{n-2}(x)\sin(x)\,dx\!} = cos n − 1 ⁡ ( x ) sin ⁡ ( x ) + ( n − 1 ) ∫ cos n − 2 ⁡ ( x ) sin 2 ⁡ ( x ) d x {\displaystyle =\cos ^{n-1}(x)\sin(x)+(n-1)\int \cos ^{n-2}(x)\sin ^{2}(x)\,dx\!} = cos n − 1 ⁡ ( x ) sin ⁡ ( x ) + ( n − 1 ) ∫ cos n − 2 ⁡ ( x ) ( 1 − cos 2 ⁡ ( x ) ) d x {\displaystyle =\cos ^{n-1}(x)\sin(x)+(n-1)\int \cos ^{n-2}(x)(1-\cos ^{2}(x))\,dx\!} = cos n − 1 ⁡ ( x ) sin ⁡ ( x ) + ( n − 1 ) ∫ cos n − 2 ⁡ ( x ) d x − ( n − 1 ) ∫ cos n ⁡ ( x ) d x {\displaystyle =\cos ^{n-1}(x)\sin(x)+(n-1)\int \cos ^{n-2}(x)\,dx-(n-1)\int \cos ^{n}(x)\,dx\!} = cos n − 1 ⁡ ( x ) sin ⁡ ( x ) + ( n − 1 ) I n − 2 − ( n − 1 ) I n {\displaystyle =\cos ^{n-1}(x)\sin(x)+(n-1)I_{n-2}-(n-1)I_{n}\,} I n + ( n − 1 ) I n = cos n − 1 ⁡ ( x ) sin ⁡ ( x ) + ( n − 1 ) I n − 2 {\displaystyle I_{n}+(n-1)I_{n}=\cos ^{n-1}(x)\sin(x)+(n-1)I_{n-2}\,} n I n = cos n − 1 ⁡ ( x ) sin ⁡ ( x ) + ( n − 1 ) I n − 2 {\displaystyle nI_{n}=\cos ^{n-1}(x)\sin(x)+(n-1)I_{n-2}\,} I n = 1 n cos n − 1 ⁡ ( x ) sin ⁡ ( x ) + n − 1 n I n − 2 {\displaystyle I_{n}={\frac {1}{n}}\cos ^{n-1}(x)\sin(x)+{\frac {n-1}{n}}I_{n-2}\,} 因此 ∫ cos n ⁡ ( x ) d x {\displaystyle \int \cos ^{n}(x)\,dx\!} 可表示为： ∫ cos n ⁡ ( x ) d x = 1 n cos n − 1 ⁡ ( x ) sin ⁡ ( x ) + n − 1 n ∫ cos n − 2 ⁡ ( x ) d x {\displaystyle \int \cos ^{n}(x)\,dx={\frac {1}{n}}\cos ^{n-1}(x)\sin(x)+{\frac {n-1}{n}}\int \cos ^{n-2}(x)\,dx\!} 将n=5代入，可得： n = 5 {\displaystyle n=5\,} ： I 5 = 1 5 cos 4 ⁡ ( x ) sin ⁡ ( x ) + 4 5 I 3 {\displaystyle I_{5}={\tfrac {1}{5}}\cos ^{4}(x)\sin(x)+{\tfrac {4}{5}}I_{3}\,} n = 3 {\displaystyle n=3\,} ： I 3 = 1 3 cos 2 ⁡ ( x ) sin ⁡ ( x ) + 2 3 I 1 {\displaystyle I_{3}={\tfrac {1}{3}}\cos ^{2}(x)\sin(x)+{\tfrac {2}{3}}I_{1}\,} ∵ I 1 = ∫ cos ⁡ ( x ) d x = sin ⁡ ( x ) + C 1 {\displaystyle \because I_{1}=\int \cos(x)\,dx=\sin(x)+C_{1}\,} ∴ I 3 = 1 3 cos 2 ⁡ ( x ) sin ⁡ ( x ) + 2 3 sin ⁡ ( x ) + C 2 {\displaystyle \therefore I_{3}={\tfrac {1}{3}}\cos ^{2}(x)\sin(x)+{\tfrac {2}{3}}\sin(x)+C_{2}\,} ， C 2 = 2 3 C 1 {\displaystyle C_{2}={\tfrac {2}{3}}C_{1}\,} I 5 = 1 5 cos 4 ⁡ ( x ) sin ⁡ ( x ) + 4 5 [ 1 3 cos 2 ⁡ ( x ) sin ⁡ ( x ) + 2 3 sin ⁡ ( x ) ] + C {\displaystyle I_{5}={\frac {1}{5}}\cos ^{4}(x)\sin(x)+{\frac {4}{5}}\left[{\frac {1}{3}}\cos ^{2}(x)\sin(x)+{\frac {2}{3}}\sin(x)\right]+C\,} ，C为常数 除了上述的 ∫ cos n ⁡ ( x ) d x {\displaystyle \int \cos ^{n}(x)\,dx\!} 外，常见的降次公式还有： ∫ sin n ⁡ ( x ) d x = − 1 n sin n − 1 ⁡ ( x ) cos ⁡ ( x ) + n − 1 n ∫ sin n − 2 ⁡ ( x ) d x {\displaystyle \int \sin ^{n}(x)\,dx=-{\frac {1}{n}}\sin ^{n-1}(x)\cos(x)+{\frac {n-1}{n}}\int \sin ^{n-2}(x)\,dx\!} ∫ tan n ⁡ ( x ) d x = 1 n − 1 tan n − 1 ⁡ ( x ) − ∫ tan n − 2 ⁡ ( x ) d x {\displaystyle \int \tan ^{n}(x)\,dx={\frac {1}{n-1}}\tan ^{n-1}(x)-\int \tan ^{n-2}(x)\,dx\!} ∫ ( ln ⁡ ( x ) ) n d x = x ( ln ⁡ ( x ) ) n − n ∫ ( ln ⁡ ( x ) ) n − 1 d x {\displaystyle \int (\ln(x))^{n}\,dx=x(\ln(x))^{n}-n\int (\ln(x))^{n-1}\,dx\!}