Question

Use the chain rule to prove the formula for integration by substitution:

$\int f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx=f\left(u\right(x\left)\right)+C.$

Short answer

After using the chain rule for integration by substitution its is proved that$\int f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx=f\left(u\right(x\left)\right)+C.$

Step-by-step solution

Step 1. Given Information 

Use the chain rule to prove the formula for integration by substitution:

$\int f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx=f\left(u\right(x\left)\right)+C.$

Step 2. To solve taking the left hand side integral.

$y=\int f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx$

Let

$$\begin{gathered} t=u\left(x\right) \\ \frac{dt}{dx}=u\text{'}\left(x\right) \\ dt=u\text{'}\left(x\right)dx \end{gathered}$$

Step 3. Now the integral after substitution.

$$\begin{gathered} \int f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx=\int f\text{'}\left(t\right)dt \\ \int f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx=f\left(t\right)+C \end{gathered}$$

Substituting the value of $t$.

$\int f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx=f\left(u\right(x\left)\right)+C$