Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       
(a) ∫¹₋₁ 𝓍ƒ(𝓍²) d𝓍

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(e) Find the value of s such that H (𝓍) = sH(―𝓍)

11-14. {Use of Tech} Compute the absolute and relative errors in using c to approximate x.

12. x = √2; c = 1.414

118. Two worthy integrals

b. Let f be any positive continuous function on the interval [0, π/2]. Evaluate

∫ from 0 to π/2 of [f(cos x) / (f(cos x) + f(sin x))] dx.

(Hint: Use the identity cos(π/2 − x) = sin x.)

(Source: Mathematics Magazine 81, 2, Apr 2008)

125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.

a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       

(b) ∫²₋₂ 𝓍²ƒ(𝓍³) d𝓍

Find ${\int }_0^{6}f\left(x\right)dx$ given the graph of $y=f\left(x\right)$.

Express the following limit as a definite integral on the interval $[0,10]$.

${\lim }_{n\to \infty }{\sum }_{k=1}^n{(x_k^{\ast }-3)}^2\Delta x$

Given the following definite integral of the function $f\left(x\right)=3x^2-2x$, write the simplified integral:

$-\int_4^0\!f\left(x\right)\,dx$