Question

Compute the integral (a) int sin (pi x) dx by letting u = pi x. (b) int e^(x/2) dx by letting u = x/2. Show that you got the correct answer by differentiating the computed function.

Answer 1

Answer:

(a) $\int sin(\pi x) dx$$=-\frac {cos \pi x}{\pi}+c$

(b)$\int e^\frac{x}{2} dx$ $=2e^\frac{x}{2}+c$

Step-by-step explanation:

(a)

$\int sin(\pi x) dx$

Let u = π x

differentiating with respect to x

du = π dx

$\Rightarrow dx=\frac{du}{\pi}$

Putting the value of x and dx

$=\int sin u \frac{du}{\pi}$

$=\frac{-cos u}{\pi}+c$    [ c is an arbitrary constant]

Now putting the value of u

$=-\frac {cos \pi x}{\pi}+c$  

(b)

$\int e^\frac{x}{2} dx$

Let $u=\frac{x}{2}$

differentiating with respect to x

$du= \frac{1}{2} dx$

2du = dx

Putting the value of x and dx

=$\int e^u.2.du$

=$2e^u +c$

Now putting the value of u

$=2e^\frac{x}{2}+c$       [ c is an arbitrary constant]