(十三) 含有指数函数的积分 (其中$a>0$)

$$ 122.\,\int\!\! a^x dx = \frac{1}{\ln a} a^x +C $$

$$ 123.\,\int\!\! e^{ax}dx = \frac{1}{a} e ^{ax}+C $$

$$ 124.\,\int\!\! xe^{ax}dx=\frac{1}{a^2}(ax-1)e^{ax}+c $$

$$ 125.\,\int\!\! x^n e^{ax}dx=\frac{1}{a}x^n e^{ax}- \frac{n}{a}\int\!\! x^{n-1} e^{ax}dx $$

$$ 126.\,\int\!\! xa^x dx= \frac{x}{\ln a}a^x - \frac{1}{(\ln a)^2}a^x+C $$

$$ 127.\,\int\!\! x^n a^x dx=\frac{1}{\ln a}x^na^x - \frac{n}{\ln a}\int\!\! x^{n-1} a^x dx $$

$$ 128.\,\int\!\! e^{ax} \sin bx dx = \frac{1}{a^2+b^2}e^{ax}(a\sin bx -b \cos bx)+C $$

$$ 129.\,\int\!\! e^{ax}\cos bx dx = \frac{1}{a^2+b^2} e^{ax} (b \sin bx + a \cos bx) +C $$

$$ 130.\,\int\!\! e^{ax}\sin ^n bx dx = \frac{1}{a^2+b^2n^2}e^{ax}\sin ^{n-1}bx (a \sin bx - nb \cos bx) $$

$$ \qquad \qquad \qquad \qquad \qquad+ \frac{n(n-1)b^2}{a^2+b^2n^2}\int\!\! e^{ax}\sin ^{n-2}bx dx $$

$$ 131.\,\int\!\! e^{ax} \cos ^n bx dx = \frac{1}{a^2+b^2n^2} e^{ax} \cos ^{n-1} bx (a\cos bx + nb \sin bx) $$

$$ \qquad \qquad \qquad \qquad \qquad + \frac{n(n-1)b^2}{a^2+b^2n^2} \int\!\! e^{ax} \cos ^{n-2}bx dx $$

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