Evaluate the Integral

Problem

(∫_^)(1/√(,2*x)*d(x))

Solution

1. Rewrite the integrand using exponent notation to make it easier to integrate.

1/√(,2*x)=1/(√(,2)⋅√(,x))

1/(√(,2)⋅√(,x))=1/√(,2)*x(−1/2)

2. Factor out the constant from the integral.

(∫_^)(1/√(,2)*x(−1/2)*d(x))=1/√(,2)*(∫_^)(x(−1/2)*d(x))

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

1/√(,2)*(∫_^)(x(−1/2)*d(x))=1/√(,2)⋅(x(1/2))/(1/2)+C

4. Simplify the expression by multiplying by the reciprocal of the denominator.

1/√(,2)⋅2*x(1/2)+C=(2√(,x))/√(,2)+C

5. Rationalize or simplify the constant coefficient.

2/√(,2)=√(,2)

√(,2)⋅√(,x)+C=√(,2*x)+C

Final Answer

(∫_^)(1/√(,2*x)*d(x))=√(,2*x)+C

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