I=∫ e ^ ax .sin( bx+c #10;) #160; dx =∫sin( bx+c ) . e ^ ax dx #160; #10; =sin( bx+c ) ∫( e ^ ax dx ) #160; #10; ∫( d dx #160; ) ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Integration by Parts

∫ eax.sinbx+c...

Question

$\int e^{a x} . sin (b x + c) d x$

Open in App

Solution

$I = \int e^{a x} . sin (b x + c) d x = \int sin (b x + c) . e^{a x} d x = sin (b x + c) \int (e^{a x} d x) - \int (\frac{d}{d x}) sin (b x + c) \int e^{a x} d x \int d x = sin (b x + c) \frac{e^{a x}}{a} - \int b . cos (b x + c) . \frac{e^{a x}}{a} d x = sin (b x + c) e^{a x} - \frac{b}{a} \int cos (b x + c) . e^{a x} . d x$

Again using integration by parts

$= sin (b x + c) e^{a x} - \frac{b}{a} (cos (b x + c) (\int e^{a x} d x) - \int (\frac{d}{d x} cos (b + c) \int e^{a x} d x) d x) y = e^{a x} sin (b x + c) - \frac{b}{a} (cos (b x + c) \frac{e^{a x}}{a} - \int - b . sin (b x + c) \frac{e^{a x}}{a} d x) = e^{a x} sin (b x + c) - \frac{b}{a^{2}} cos (b x + c) e^{a x} - \frac{b^{2}}{a^{2}} \int sin (b x + c) e^{a x} d x I = e^{a x} sin (b x + c) - \frac{b}{a^{2}} cos (b x + c) e^{a x} - \frac{b^{2}}{a^{2}} I I + \frac{b^{2}}{a^{2}} I = sin (b x + c) e^{a x} - \frac{b}{a^{2}} cos (b x + c) e^{a x} \Rightarrow I = \frac{e^{a x}}{(1 + \frac{b^{2}}{a^{2}})} (sin (b x + c) - \frac{b}{a^{2}} cos (b x + c))$

Suggest Corrections