Answer:
(a) [tex]\int sin(\pi x) dx[/tex][tex]=-\frac {cos \pi x}{\pi}+c[/tex]
(b)[tex]\int e^\frac{x}{2} dx[/tex] [tex]=2e^\frac{x}{2}+c[/tex]
Step-by-step explanation:
(a)
[tex]\int sin(\pi x) dx[/tex]
Let u = π x
differentiating with respect to x
du = π dx
[tex]\Rightarrow dx=\frac{du}{\pi}[/tex]
Putting the value of x and dx
[tex]=\int sin u \frac{du}{\pi}[/tex]
[tex]=\frac{-cos u}{\pi}+c[/tex] [ c is an arbitrary constant]
Now putting the value of u
[tex]=-\frac {cos \pi x}{\pi}+c[/tex]
(b)
[tex]\int e^\frac{x}{2} dx[/tex]
Let [tex]u=\frac{x}{2}[/tex]
differentiating with respect to x
[tex]du= \frac{1}{2} dx[/tex]
2du = dx
Putting the value of x and dx
=[tex]\int e^u.2.du[/tex]
=[tex]2e^u +c[/tex]
Now putting the value of u
[tex]=2e^\frac{x}{2}+c[/tex] [ c is an arbitrary constant]
