Evaluate the Integral

Problem

(∫_^)(x√(,4−x)*d(x))

Solution

1. Identify the substitution to simplify the radicand. Let u=4−x

2. Differentiate the substitution to find the relationship between d(x) and d(u)

u=4−x

d(u)/d(x)=−1

d(x)=−d(u)

3. Solve for x in terms of u to substitute the x term outside the square root.

x=4−u

4. Substitute the expressions for x √(,4−x) and d(x) into the integral.

(∫_^)((4−u)√(,u)*(−d(u)))

5. Distribute the negative sign and the √(,u) (which is u(1/2) into the parentheses.

(∫_^)((u−4)*u(1/2)*d(u))

(∫_^)((u(3/2)−4*u(1/2))*d(u))

6. Integrate term by term using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

(u(5/2))/(5/2)−4(u(3/2))/(3/2)+C

2/5*u(5/2)−8/3*u(3/2)+C

7. Back-substitute u=4−x to return to the original variable.

2/5*(4−x)(5/2)−8/3*(4−x)(3/2)+C

Final Answer

(∫_^)(x√(,4−x)*d(x))=2/5*(4−x)(5/2)−8/3*(4−x)(3/2)+C

Want more problems? Check here!