Question

What is the answer of the question?​

Answer 1

A. 2

Explanation:

`\int_0^1 \: 5x \sqrt x \: dx \\ \\ = \int_0^1 \: 5x . \: {x}^{ (1)/(2) } \: dx \\ \\ = 5\int_0^1 \: {x}^{ (3)/(2) } \: dx \\ \\ = 5 \bigg( \frac{ {x}^{ (3)/(2) + 1 } }{ (3)/(2) + 1 } \bigg)_0^1\\ \\ = 5 \bigg( \frac{ {x}^{ (5)/(2) } }{ (5)/(2)} \bigg)_0^1 \\ \\ = 5 * (2)/(5) \bigg({x}^{ (5)/(2) } \bigg)_0^1 \\ \\ = 2 \bigg( {1}^{ (5)/(2) } - {0}^{ (5)/(2) } \bigg) \\ \\ = 2(1 - 0) \\ \\ = 2`

Answer 2

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 }}`