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Pattern - 15 of Indefinite Integration

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⋆ Pattern – 15

1) ∫ $e^{ax}$ sin (bx + k) dx

$=\ \frac{e^{ax}}{a^2+b^2}\left[a\sin{\left(bx+k\right)}-b\cos{\left(bx+k\right)}\right]+c$

$=\ \frac{e^{ax}}{\sqrt{a^2+b^2}}\sin{\left(bx+k-\ \alpha\right)}+c$

where, $\alpha=\tan^{-1}{\frac{b}{a}}$

2) ∫ $e^{ax}$ cos ( bx + k ) dx

$=\ \frac{e^{ax}}{a^2+b^2}\ \left[a\cos{\left(bx+k\right)}+b\sin{\left(bx+k\right)}\right]+c$

$=\ \frac{e^{ax}}{\sqrt{a^2+b^2}}\cos{\left(bx+k-\ \alpha\right)}+c$

where, $\alpha=\tan^{-1}{\frac{b}{a}}$

1) I = ∫ $3^x$ sin 2x dx

$3^x=e^{\log_e{3^x}\ }$

$=e^{\ \ x\log{3}}$

$\therefore\ \ I=\ \int e^{x\bullet\log{3}}\times \sin{2x}\ dx$

$=\ \frac{e^{x\log{2}\ }}{\left(\log{3}\right)^2+4}\ \left[\log{3}\sin{2x}-2\cos{2x}\right]+c$

$=\ \frac{3^x}{\left(\log{3}\right)^2+4}\ \left[\log{3}\times \sin{2x}-2\cos{2x}\right]+c$

$\left(Here,\ 3^x=e^{\log_e{3^x}}=e^{\left(x\log{3}\right)}\right)$

$=\ \int e^{x\log{3}}\sin{2x\ }\ dx$

$=\ \frac{e^{x\log{3}}}{\sqrt{\left(\log{3}\right)^2+4}}\sin{\left(2x-\ \alpha\right)}+c$

$\left(where,\ \alpha=\tan^{-1}{\frac{2}{\log{3}}}\right)$

$=\ \frac{3^x}{\sqrt{\left(\log{3}\right)^2+4}}\sin{\left(2x-\ \alpha\right)}+c$

$\left(where,\ \alpha=\tan^{-1}{\frac{2}{\log{3}}}\right)$