# Evaluate the following problems using second fundamental theorem: $\int\limits_{0}^{\large\frac{\pi}{2}} e^{3x}\cos x dx$

Toolbox:
• If $F(x)=\int \limits_a^x f(t)dt$ then $\int \limits_a^b f(x) dx=F(b)-F(a)$
$\int \limits_0^{\large\frac{\pi}{2}} e^{3x} \cos x dx$
Step 1:
Let $I= \int e^{3x} \cos x=\int udv$
Where $u= \cos x\;and\; dv=e^{3x}dx$
$du=-\sin x \;dx\; and \;v=\large\frac{e^{3x}}{3}$
$I= uv-\int vdu$
$\quad= \cos x \large\frac{e^{3x}}{3}+\int \large\frac {e^3x}{3}$$\sin x dx Step 2: Let I_1= \large\frac{1}{3} \int e^{3x}$$ \sin x dx =\int udv$
Where $u= \sin x,\; dv=e^{3x}dx$
$du=\cos x \;dx,\;v=\large\frac{e^{3x}}{3}$
$I_1= uv-\int vdu$
$\quad= \large\frac{e^{3x}}{9}$$\sin x -\large\frac{1}{9} \int$$ e^3x \cos x dx$
$\quad= \large\frac{e^{3x}}{9}$$\sin x -\large\frac{1}{9}$$ I$
Step 3:
$\therefore I=\cos x \large\frac{e^{3x}}{3}+\frac{e^{3x}}{9}$$\sin x-\large\frac{1}{9}$$ I$
$\large\frac{10}{9}$$I=\large\frac{e^{3x}}{3}$$ [\cos x+\large\frac{\sin x}{3}]$
$I=\large\frac{3}{10}$$e^{3x} [\cos x+\large\frac{\sin x}{3}] \int \limits_0^{\large\frac{\pi}{2}} e^{3x} \cos x dx=\bigg[\large\frac{3}{10}$$ e^3x [\cos x +\large\frac{\sin x}{3}]\bigg]_0^{\large\frac{\pi}{2}}$
$\qquad=\large\frac{1}{10} e^{\large\frac{3 \pi}{2}}-\large\frac{3}{10}$