Question

$\int e^{2x-3}+7^{4-3\left(x/2\right)}+\sin (3x-\frac{1}{2})+\cos (\frac{2}{5}x-2)+a^{3x+2}dx$ is

Answer 1

Let $I=\int e^{2x-3}+7^{4-3\left(x/2\right)}+\sin (3x-\frac{1}{2})+\cos (\frac{2}{5}x-2)+{\alpha }^{3x+2}dx=I_1+I_2+I_3+I_4+I_5$
Where
$I_1=\int e^{2x-3}dx=\frac{1}{2}{e}^{2x-3}$
$I_2=\int 7^{4-3\left(x/2\right)}dx$
Put $t=4-\frac{3x}{2}\Rightarrow dt=-\frac{3}{2}dx$
$I_2=-\frac{2}{3}\int 7^{t}dt=-\frac{2}{3}\frac{7^t}{\log 7}=-\frac{2}{3}\cdot 7^{4-3\left(x/2\right)}\frac{1}{\log 7}$
$I_3=\int \sin (3x-\frac{1}{2})dx=-\frac{1}{3}\cos (3x-\frac{1}{2})$
$I_4=\int \cos (\frac{2}{5}x-2)dx=\frac{5}{2}\sin (\frac{2}{5}x-2)$
$I_5=\int a^{3x+2}dx=\frac{1}{3}\frac{a^{3x+2}}{\log a}$
Therefore
$I=\frac{1}{2}{e}^{2x-3}-\frac{2}{3}\cdot 7^{4-3\left(x/2\right)}\frac{1}{\log 7}-\frac{1}{3}\cos (3x-\frac{1}{2})+\frac{5}{2}\sin (\frac{2}{5}x-2)+\frac{1}{3}\frac{a^{3x+2}}{\log a}$