Evaluate the Integral

Problem

(∫_1^4)(5*x2+√(,x)*d(x))

Solution

1. Rewrite the integrand by expressing the square root as a power of x

(∫_1^4)(5*x2+x(1/2)*d(x))

2. Apply the power rule for integration, which states (∫_^)(xn*d(x))=(x(n+1))/(n+1) to each term.

[(5*x3)/3+(x(3/2))/(3/2)]41

3. Simplify the coefficients of the antiderivative.

[5/3*x3+2/3*x(3/2)]41

4. Substitute the upper limit of integration (x=4 into the expression.

5/3*(4)3+2/3*(4)(3/2)=5/3*(64)+2/3*(8)=320/3+16/3=336/3

5. Substitute the lower limit of integration (x=1 into the expression.

5/3*(1)3+2/3*(1)(3/2)=5/3+2/3=7/3

6. Subtract the lower limit value from the upper limit value.

336/3−7/3=329/3

Final Answer

(∫_1^4)(5*x2+√(,x)*d(x))=329/3

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