KVPY2014MathematicsDefinite Integration
Let f:[0,1] [0, ) be a continuous function such that ₀¹ f(x) d x=10 . Which of the following statements is NOT necessarily true?
Options
- A₀¹ e^ -x f(x) d x 10
- B₀¹ f(x) (1+x)² d x 10
- C-10 ₀¹ (100 x) f(x) d x 10
- D₀¹ f(x)² d x 100
Correct answer
D. ₀¹ f(x)² d x 100
Step-by-step solution
I= ₀¹ f (x² ) d x Let x²=t dx = 1 2 t dt I =1 ₀¹ f ( t ) dt 2 t 1 2 1 2 t ₀¹ f ( t ) dt 2 ₀¹ f ( t ) dt 2 t 5 l