Question

Solve in attachment....​

Answer 1

2 ( Option A )

Explanation:

The given integral to us is ,

`\longrightarrow \displaystyle \int_0^1 5x √(x)\ dx`

Here 5 is a constant so it can come out . So that,

`\longrightarrow \displaystyle I = 5 \int_0^1 x √(x)\ dx`

Now we can write √x as ,

`\longrightarrow I = \displaystyle 5 \int_0^1 x . x^{(1)/(2)} \ dx`

Simplify ,

`\longrightarrow I = 5 \displaystyle \int_0^1 x^{(3)/(2)}\ dx`

By Power rule , the integral of x^3/2 wrt x is , 2/5x^5/2 . Therefore ,

`\longrightarrow I = 5 \bigg( (2)/(5) x^{(5)/(2)} \bigg] ^1_0 \bigg)`

On simplifying we will get ,

`\longrightarrow \underline{\underline{ I = 2 }}`

Answer 2

A)2

Explanation:

we would like to integrate the following definite Integral:

`\displaystyle \int_(0) ^(1) 5x √(x) dx`

use constant integration rule which yields:

`\displaystyle 5\int_(0) ^(1) x √(x) dx`

notice that we can rewrite √x using Law of exponent therefore we obtain:

`\displaystyle 5\int_(0) ^(1) x \cdot {x}^(1/2) dx`

once again use law of exponent which yields:

`\displaystyle 5\int_(0) ^(1) {x}^{ (3)/(2) } dx`

use exponent integration rule which yields;

`\displaystyle 5 \left( \frac{{x}^{ (3)/(2) + 1 } }{ (3)/(2) + 1} \right) \bigg| _(0) ^(1)`

simplify which yields:

`\displaystyle 2 {x}^(2) √(x) \bigg| _(0) ^(1)`

recall fundamental theorem:

`\displaystyle 2 ( {1}^(2)) (√(1) ) - 2( {0}^(2) )( √(0))`

simplify:

`\displaystyle 2`

hence

our answer is A