Question

I’m trying to help my friend solve this question. Please help

Answer 1

The
`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx` integral has a calculated value of 5.276

How to evaluate the integral

From the question, we have the following parameters that can be used in our computation:

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx`

This can be expressed as

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx = \int\limits^3_0 {[2^\frac x2]} \, dx`

Let u = x/2

So, we have

du = dx/2

And as such

dx = 2 du

By substitution, we have

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx = \int\limits^3_0 {[2^u]} \, 2 * du`

Factor out 2

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx = 2\int\limits^3_0 {[2^u]} \, du`

Applying the exponential rule of integration, we have

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx = (2* 2^u)/(\ln(2)) \limits^3_0`

Recall that

u = x/2

So, we have

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx = (2* 2^\frac x2)/(\ln(2)) \limits^3_0`

Expand

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx = (2* 2^\frac 32)/(\ln(2)) - (2* 2^\frac 02)/(\ln(2))`

This gives

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx = (2^\frac 52)/(\ln(2)) - (2)/(\ln(2))`

This gives

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx = (2^\frac 52 - 2)/(\ln(2))`

Evaluate

`\int\limits^3_0 {[(\sqrt 2)^x]} \, dx = 5.276`

Hence, the integral has a value of 5.276