Evaluation Of Limits

Let $I%20%3D%20%5Cint_0%5E%7B%5Cfrac%7B%5Cpi%5E2%7D%7B4%7D%7D%20%5Cfrac%7B%5Csin%5Csqrt%7Bx%7D%7D%7B%5Csqrt%7Bx%7D%7D%20%5C%2C%20dx$ .

Using substitution, let $t%20%3D%20%5Csqrt%7Bx%7D$ . Then, $x%20%3D%20t%5E2%2C%20%5Cquad%20dx%20%3D%202t%20%5C%2C%20dt%2C%20%5Cquad%20%5Csqrt%7Bx%7D%20%3D%20t.$ Substitute into the integral: $I%20%3D%20%5Cint_0%5E%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%20%5Cfrac%7B%5Csin%20t%7D%7Bt%7D%20%5Ccdot%202t%20%5C%2C%20dt%20%3D%202%20%5Cint_0%5E%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%20%5Csin%20t%20%5C%2C%20dt.$

Simplify: $I%20%3D%202%20%5Cleft%5B%20-%5Ccos%20t%20%5Cright%5D_0%5E%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D.$ Evaluate: $I%20%3D%202%20%5Cleft%5B%20-%5Ccos%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%20%2B%20%5Ccos(0)%20%5Cright%5D%20%3D%202%20%5Cleft%5B%200%20%2B%201%20%5Cright%5D%20%3D%202.$

Answer: $%5Cint_0%5E%7B%5Cfrac%7B%5Cpi%5E2%7D%7B4%7D%7D%20%5Cfrac%7B%5Csin%5Csqrt%7Bx%7D%7D%7B%5Csqrt%7Bx%7D%7D%20%5C%2C%20dx%20%3D%202.$ \bigskip

Let $I%20%3D%20%5Cint%20x%20%5Csqrt%7B1%20%2B%202x%7D%20%5C%2C%20dx$ .

Using substitution, let $u%20%3D%201%20%2B%202x$ . Then, $du%20%3D%202%20%5C%2C%20dx%20%5Cquad%20%5Ctext%7Band%7D%20%5Cquad%20x%20%3D%20%5Cfrac%7Bu%20-%201%7D%7B2%7D.$

Substitute into the integral: $I%20%3D%20%5Cint%20%5Cfrac%7Bu%20-%201%7D%7B2%7D%20%5Csqrt%7Bu%7D%20%5Ccdot%20%5Cfrac%7B1%7D%7B2%7D%20%5C%2C%20du%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%5Cint%20(u%20-%201)%20u%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%5C%2C%20du.$

Simplify: $I%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%5Cint%20%5Cleft(u%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20-%20u%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Cright)%20%5C%2C%20du.$

Split the integral: $I%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%5Cleft(%20%5Cint%20u%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%5C%2C%20du%20-%20%5Cint%20u%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%5C%2C%20du%20%5Cright).$

Integrate each term: $%5Cint%20u%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%5C%2C%20du%20%3D%20%5Cfrac%7B2%7D%7B5%7D%20u%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%2C%20%5Cquad%20%5Cint%20u%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%5C%2C%20du%20%3D%20%5Cfrac%7B2%7D%7B3%7D%20u%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D.$ Substitute back: $I%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%5Cleft(%5Cfrac%7B2%7D%7B5%7D%20u%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%20-%20%5Cfrac%7B2%7D%7B3%7D%20u%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Cright).$

Simplify and substitute $u%20%3D%201%20%2B%202x$ : $I%20%3D%20%5Cfrac%7B1%7D%7B10%7D%20(1%20%2B%202x)%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%20-%20%5Cfrac%7B1%7D%7B6%7D%20(1%20%2B%202x)%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%2B%20C.$
Answer: $%5Cint%20x%20%5Csqrt%7B1%20%2B%202x%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B1%7D%7B10%7D%20(1%20%2B%202x)%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D%20-%20%5Cfrac%7B1%7D%7B6%7D%20(1%20%2B%202x)%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%2B%20C.$

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Evaluation Of Limits

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs