Evaluate the Integral

Problem

(∫_0^1)(√(3,1+7*x)*d(x))

Solution

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

(∫_0^1)((1+7*x)(1/3)*d(x))

2. Identify the substitution u=1+7*x to simplify the integrand.

u=1+7*x

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=7

d(x)=1/7*d(u)

4. Change the limits of integration from x to u

When *x=0,u=1+7*(0)=1

When *x=1,u=1+7*(1)=8

5. Substitute the new variables and limits into the integral.

(∫_1^8)(u(1/3)⋅1/7*d(u))

1/7*(∫_1^8)(u(1/3)*d(u))

6. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)

1/7*[(u(4/3))/(4/3)]81

1/7⋅3/4*[u(4/3)]81

3/28*[u(4/3)]81

7. Evaluate the expression at the upper and lower limits.

3/28*(8(4/3)−1(4/3))

8. Simplify the numerical values. Note that 8(4/3)=(√(3,8))4=2=16

3/28*(16−1)

3/28⋅15

45/28

Final Answer

(∫_0^1)(√(3,1+7*x)*d(x))=45/28

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