Find the Integral 6 square root of x

Problem

(∫_^)(6√(,x)*d(x))

Solution

1. Rewrite the radical expression using a fractional exponent to make it easier to integrate.

√(,x)=x^(1/2)

2. Apply the constant multiple rule by moving the constant factor outside the integral.

(∫_^)(6*x^(1/2)*d(x))=6*(∫_^)(x^(1/2)*d(x))

3. Apply the power rule for integration, which states that (∫_^)(x^n*d(x))=(x^(n+1))/(n+1) for n≠−1

6*(∫_^)(x^(1/2)*d(x))=6⋅(x^(1/2+1))/(1/2+1)+C

4. Simplify the exponent and the denominator.

6⋅(x^(3/2))/(3/2)+C

5. Multiply by the reciprocal of the denominator to simplify the expression.

6⋅2/3*x^(3/2)+C

6. Calculate the final coefficient.

4*x^(3/2)+C

Final Answer

(∫_^)(6√(,x)*d(x))=4*x^(3/2)+C

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