Trouble integrating $e^{ax}\cos bx$ and $e^{ax}\sin bx$

Question

I have to integrate the following indefinite integrals $$\int e^{ax}\sin bxdx~~;~\int e^{ax}\cos bx$$ The procedure I used is the same for both integrals:

• Make the change of variables $bx=t$

• Use partial integration twice with $dv=e^{at/b}dt$ until the initial integral reappears and then solve.

This is the full procedure for the first integral. For the 2nd integral the result I obtain is the same except for the signs. $$\int e^{ax}\sin bxdx$$ Change of variables $bx=t$ $$\frac{1}{b}\int e^{at/b}\sin tdt$$ Partial integration $$dv=e^{at/b}dt~;~v=\frac{b}{a}e^{at/b}~;~u=\sin t~;~du=\cos tdt$$ Which leads to $$\frac{1}{a}e^{at/b}\sin t-\frac{1}{a}\int e^{at/b}\cos tdt$$ Partial integration again $$dv=e^{at/b}~;~v=\frac{b}{a}e^{at/b}~;~u=\cos t~;~du=-\sin tdt$$ Which leads to $$\frac{1}{b}I=\frac{1}{a}e^{at/b}\sin t-\frac{b}{a^2}e^{at/b}\cos t-\frac{b}{a^2}\int e^{at/b}\sin tdt$$ $$\frac{a^2+b^2}{ba^2}I=\frac{1}{a}e^{at/b}\sin t-\frac{b}{a^2}e^{at/b}\cos t$$ $$(a^2+b^2)I=bae^{at/b}\sin t-b^2e^{at/b}\cos t$$ $$I=\frac{ba}{a^2+b^2}e^{at/b}\sin t-\frac{b^2}{a^2+b^2}e^{at/b}\cos t$$ $$I=\frac{ba}{a^2+b^2}e^{ax}\sin bx-\frac{b^2}{a^2+b^2}e^{ax}\cos bx+C$$

The correct result is the one I obtained but divided by $b$ and for the love of God I can't figure out where I made the mistake.

Answer 1

I prefer to write this using Euler's formula (exponentials are so much easier) \begin{align} \int e^{x(a+ i b)} \, dx & = \frac{1}{a + i b} e^{x(a + ib)} \\ & = \frac{a - ib}{a^2+b^2} e^{x(a+ib)}. \end{align}

Then you can just read off $$\int e^{ax} \cos(bx) = \frac{e^{ax}}{a^2+b^2}[a \cos(bx) + b \sin(bx)]$$ and $$\int e^{ax} \sin(bx) = \frac{e^{ax}}{a^2+b^2}[a \sin(bx) - b \cos(bx)]$$

Answer 2

\begin{align*} I &= \int e^{ax} \cos bx \, dx \\ J &= \int e^{ax} \sin bx \, dx \\ I &= \frac{e^{ax}}{a} \cos bx-\int \frac{e^{ax}}{a} d(\cos bx) \\ &= \frac{e^{ax}}{a} \cos bx+\frac{b}{a}\int e^{ax} \sin bx \, dx \\ I &= \frac{e^{ax}}{a} \cos bx+\frac{b}{a} J+C_1 \tag{1} \\ J &= \frac{e^{ax}}{a} \sin bx-\int \frac{e^{ax}}{a} d(\sin bx) \\ &= \frac{e^{ax}}{a} \sin bx-\frac{b}{a}\int e^{ax} \cos bx \, dx \\ J &= \frac{e^{ax}}{a} \sin bx-\frac{b}{a} I+C_2 \tag{2} \\ \end{align*}

Solving $I$ and $J$ gives the answers.

Answer 3

Let, $I=\displaystyle\int{e^{ax}\sin{bx}}\ dx$

now by applying by parts rule \begin{align*} I&= e^{ax}\int \sin{bx}\ dx-\int\left[\frac{d}{dx}(e^{ax})\cdot\int\sin{bx}\ dx\right]\ dx\\ &= e^{ax}\left(-\frac{\cos{bx}}{b}\right)-\int{ae^{ax}}\cdot\left(-\frac{\cos{bx}}{b}\right)\ dx\\ &=-\frac{e^{ax}\cos{bx}}{b}+\frac{a}{b}\int{e^{ax}\cos{bx}}\ dx\\ &=-\frac{e^{ax}\cos{bx}}{b}+\frac{a}{b}\left[e^{ax}\int\cos{bx}\ dx-\int\left\{\frac{d}{dx}(e^{ax})\cdot\int\cos{bx}\ dx\right\}dx\right]+k_1\\ &=-\frac{e^{ax}\cos{bx}}{b}+\frac{a}{b}e^{ax}\cdot\frac{\sin{bx}}{b}-\frac{a}{b}\int ae^{ax}\cdot\frac{\sin{bx}}{b}\ dx+k_1\\ or,\ I&=-\frac{e^{ax}\cos{bx}}{b}+\frac{a}{b^2}e^{ax}\sin{bx}-\frac{a^2}{b^2}I+k_1\\ or,\ I+\frac{a^2}{b^2}I&=\frac{e^{ax}}{b^2}\left(a\sin{bx}-b\cos{bx}\right)+k_1\\ or,\ I&=\frac{e^{ax}}{a^2+b^2}\left(a\sin{bx}-b\cos{bx}\right)+k \end{align*}