Chapter 5

Evaluate the indefinite integral. \( \displaystyle \int y^2 (4 - y^3)^{2/3} \, dy \)

Find the general indefinite integral. \( \displaystyle \int \biggl( \frac{1 + r}{r} \biggr)^2 \, dr \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle h(x) = \int^{\sqrt{x}}_1 \frac{z^2}{z^4 + 1} \,dz \)

Evaluate the indefinite integral. \( \displaystyle \int \cos^3 \theta \sin \theta \, d\theta \)

Find the general indefinite integral. \( \displaystyle \int (2 + \tan^2 \theta)\, d\theta \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle y = \int^{3x + 2}_1 \frac{t}{1 + t^3} \,dt \)

Use a calculator or computer to make a table of values of right Riemann sums \( R_n \) for the integral \( \int^{\pi}_0 \sin x \, dx \) with \( n \) = 5, 10, 50, and 100. What value do these numbers appear to be approaching?

Evaluate the indefinite integral. \( \displaystyle \int e^{-5r} \, dr \)

Find the general indefinite integral. \( \displaystyle \int \sec t (\sec t + \tan t)\, dt \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle y = \int^{x^4}_0 \cos^2 \theta \,d\theta \)

Use a calculator or computer to make a table of values of left and right Riemann sums \( L_n \) and \( R_n \) for the integral \( \displaystyle \int^2_0 e^{-x^2} \, dx \) with \( n \) = 5, 10, 50, and 100. Between what two numbers must the value of the integral lie? Can you make a similar statement for the integral \( \displaystyle \int^2_{-1} e^{-x^2} \, dx \)? Explain.

Evaluate the indefinite integral. \( \displaystyle \int \frac{e^u}{(1 - e^u)^2} \, du \)

Find the general indefinite integral. \( \displaystyle \int 2^t (1 + 5^t)\, dt \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle y = \int^{\pi/4}_{\sqrt{x}} \theta \tan \theta \,d\theta \)

Express the limit as a definite integral on the given interval. \( \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \frac{e^{x_i}}{1 + x_i} \, \Delta x \), [0, 1]

Evaluate the indefinite integral. \( \displaystyle \int \frac{\sin \sqrt{x}}{\sqrt{x}} \, dx \)

Find the general indefinite integral. \( \displaystyle \int \frac{\sin 2x}{\sin x}\, dx \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle y = \int^1_{\sin x} \sqrt{1 + t^2} \,dt \)

Express the limit as a definite integral on the given interval. \( \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n x_i \sqrt{1 + x_i^3} \, \Delta x \), [2, 5]

Evaluate the indefinite integral. \( \displaystyle \int \frac{a + bx^2}{\sqrt{3ax + bx^3}} \, dx \)

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. \( \displaystyle \int \biggl( \cos x + \frac{1}{2}x \biggr) \,dx \)

Evaluate the integral. \( \displaystyle \int^3_1 (x^2 + 2x - 4) \,dx \)

Express the limit as a definite integral on the given interval. \( \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n [5(x_i^*)^3 - 4x_i^*] \, \Delta x \), [2, 7]

In someone affected with measles, the virus level \( N \) (measured in number of infected cells per mL of blood plasma) reaches a peak density at about \( t = 12 \) days (when a rash appears) and then decreases fairly rapidly as a result of immune response. The area under the graph of \( N(t) \) from \( t = 0 \) to \( t = 12 \) (as shown in the figure) is equal to the total amount of infection needed to develop symptoms (measured in density of infected cells x time). The function \( N \) has been modeled by the function $$ f(t) = -t(t - 21)(t + 1) $$ Use this model with six subintervals and their midpoints to estimate the total amount of infection needed to develop symptoms of measles.

Evaluate the indefinite integral. \( \displaystyle \int \frac{z^2}{z^3 + 1} \, dz \)

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. \( \displaystyle \int (e^x - 2x^2) \,dx \)

Evaluate the integral. \( \displaystyle \int^1_{-1} x^{100} \,dx \)

Express the limit as a definite integral on the given interval. \( \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \frac{x_i^*}{(x_i^*)^2 + 4} \, \Delta x \), [1, 3]

Evaluate the indefinite integral. \( \displaystyle \int \frac{(\ln x)^2}{x} \, dx \)

Evaluate the integral. \( \displaystyle \int^3_{-2} (x^2 - 3) \,dx \)

Evaluate the integral. \( \displaystyle \int^2_0 \biggl(\frac{4}{5}t^3 - \frac{3}{4}t^2 + \frac{2}{5}t \biggr) \,dt \)

Evaluate the indefinite integral. \( \displaystyle \int \sin x \sin(\cos x) \, dx \)

Evaluate the integral. \( \displaystyle \int^2_{1} (4x^3 - 3x^2 + 2x) \,dx \)

Evaluate the integral. \( \displaystyle \int^1_0 (1 - 8v^3 + 16v^7) \,dv \)

Evaluate the indefinite integral. \( \displaystyle \int \sec^2 \theta \tan^3 \theta \, d\theta \)

Evaluate the integral. \( \displaystyle \int^0_{-2} \biggl( \frac{1}{2}t^4 + \frac{1}{4}t^3 - t \biggr) \,dt \)

Evaluate the integral. \( \displaystyle \int^9_1 \sqrt{x} \,dx \)

Evaluate the indefinite integral. \( \displaystyle \int x \sqrt{x + 2} \, dx \)

Evaluate the integral. \( \displaystyle \int^{3}_{0} (1 + 6w^2 - 10w^4) \,dw \)

Evaluate the integral. \( \displaystyle \int^8_1 x^{-2/3} \,dx \)

Determine a region whose area is equal to the given limit. Do not evaluate the limit. \( \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{3}{n} \sqrt{1 +\frac{3i}{n}} \)

Evaluate the indefinite integral. \( \displaystyle \int e^x \sqrt{1 + e^x} \, dx \)

Evaluate the integral. \( \displaystyle \int^{2}_{0} (2x - 3)(4x^2 + 1) \,dx \)

Evaluate the integral. \( \displaystyle \int^{\pi}_{\pi/6} \sin \theta \,d\theta \)

Determine a region whose area is equal to the given limit. Do not evaluate the limit. \( \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{\pi}{4n} \tan{\frac{i \pi}{4n}} \)

Evaluate the indefinite integral. \( \displaystyle \int \frac{dx}{ax + b} \) \( (a \neq 0) \)

Evaluate the integral. \( \displaystyle \int^{1}_{-1} t(1 - t)^2 \,dt \)

Evaluate the integral. \( \displaystyle \int^5_{-5} e \,dx \)

Evaluate the indefinite integral. \( \displaystyle \int (x^2 + 1)(x^3 + 3x)^4 \, dx \)

Evaluate the integral. \( \displaystyle \int^{\pi}_{0} (5e^x + 3\sin x) \,dx \)