29. \( \int_{-1}^{0} e^{x} d x \) 30. \( \int_{-1}^{0} e^{-x} d \) \( \begin{array}{ll}\text { 33. } \int_{0}^{1} \frac{1}{1+x^{2}} d x & \text { 32. } \int_{e}^{1} \frac{1}{x} d x \\ d t] & \text { 34. } \int_{0}^{\sqrt{2} / 2} \frac{1}{\sqrt{1-x^{2}}} d x \\ \text { 35. } \int_{-1}^{8} x^{2 / 3} d x & \text { 36. } \int_{0}^{4} x^{3 / 2} d x\end{array} \)

We will evaluate each of the following integrals one by one. ──────────────────── 29. Evaluate ∫₋₁⁰ eˣ dx The antiderivative of eˣ is eˣ. Thus,   ∫₋₁⁰ eˣ dx = [eˣ]₋₁⁰ = e⁰ – e^(–1) = 1 – 1/e. ──────────────────── 30. Evaluate ∫₋₁⁰ e^(–x) dx The antiderivative of e^(–x) is –e^(–x) (since d/dx[–e^(–x)] = e^(–x)). Thus,   ∫₋₁⁰ e^(–x) dx = [–e^(–x)]₋₁⁰ = [–e⁰] – [–e^(1)] = (–1) – (–e) = e – 1. ──────────────────── 33. Evaluate ∫₀¹ 1/(1+x²) dx The antiderivative of 1/(1+x²) is arctan(x). Thus,   ∫₀¹ 1/(1+x²) dx = [arctan(x)]₀¹ = arctan(1) – arctan(0) = (π/4) – 0 = π/4. ──────────────────── 32. Evaluate ∫ₑ¹ 1/x dx The antiderivative of 1/x is ln|x|. Thus,   ∫ₑ¹ 1/x dx = [ln|x|]ₑ¹ = ln|1| – ln|e| = 0 – 1 = –1. Note: The integration limits are from e to 1 (upper limit smaller than the lower limit), which gives a negative result. ──────────────────── 34. Evaluate ∫₀^(√2⁄2) 1/√(1–x²) dx The antiderivative of 1/√(1–x²) is arcsin(x). Thus,   ∫₀^(√2⁄2) 1/√(1–x²) dx = [arcsin(x)]₀^(√2⁄2) = arcsin(√2⁄2) – arcsin(0) = (π/4) – 0 = π/4. ──────────────────── 35. Evaluate ∫₋₁⁸ x^(2/3) dx First, find the antiderivative:   ∫ x^(2/3) dx = (x^(5/3))/(5/3) + C = (3/5) x^(5/3) + C. Thus,   ∫₋₁⁸ x^(2/3) dx = [(3/5) x^(5/3)]₋₁⁸ = (3/5)[8^(5/3) – (–1)^(5/3)]. Analyze each term:  • 8^(5/3) = (cube root of 8)⁵ = 2⁵ = 32.  • (–1)^(5/3) = (cube root of –1)⁵ = (–1)⁵ = –1. Therefore,   = (3/5)[32 – (–1)] = (3/5)[33] = 99/5. ──────────────────── 36. Evaluate ∫₀⁴ x^(3/2) dx Find the antiderivative:   ∫ x^(3/2) dx = (x^(5/2))/(5/2) + C = (2/5)x^(5/2) + C. Thus,   ∫₀⁴ x^(3/2) dx = [(2/5)x^(5/2)]₀⁴ = (2/5)[4^(5/2) – 0]. Compute 4^(5/2):   4^(5/2) = (√4)⁵ = 2⁵ = 32. So,   = (2/5)*32 = 64/5. ──────────────────── Summary of Answers: 29. 1 – 1/e 30. e – 1 33. π/4 32. –1 34. π/4 35. 99/5 36. 64/5