Chapter 5

Evaluate the integral by making the given substitution. \( \displaystyle \int \cos 2x \, dx \), \( u = 2x \)

Verify by differentiation that the formula is correct. \( \displaystyle \int \frac{1}{x^2 \sqrt{1 + x^2}} \,dx = - \frac{\sqrt{1 + x^2}}{x} + C \)

Explain exactly what is meant by the statement that "differentiation and integration are inverse processes."

Evaluate the integral by making the given substitution. \( \displaystyle \int xe^{-x^2}\, dx \), \( u = -x^2 \)

Verify by differentiation that the formula is correct. \( \displaystyle \int \cos^2 x \,dx = \frac{1}{2}x + \frac{1}{4}\sin 2x+ C \)

If $$ f(x) = \cos x \hspace{10mm} 0 \le x \le 3\pi/4 $$ evaluate the Riemann sum with \( n = 6 \), taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does the Riemann sum represent? Illustrate with a diagram.

Evaluate the integral by making the given substitution. \( \displaystyle \int x^2 \sqrt{x^3 + 1} \, dx \), \( u = x^3 + 1 \)

Verify by differentiation that the formula is correct. \( \displaystyle \int \tan^2 x \,dx = \tan x - x + C \)

(a) Estimate the area under the graph of \( f(x) = 1/x \) from \( x = 1 \) to \( x =2 \) using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

Evaluate the integral by making the given substitution. \( \displaystyle \int \sin^2 \theta \cos \theta \, d\theta \), \( u = \sin \theta \)

Verify by differentiation that the formula is correct. \( \displaystyle \int x\sqrt{a + bx} \,dx = \frac{2}{15b^2}(3bx - 2a)(a + bx)^{3/2} + C \)

(a) Estimate the area under the graph of \( f(x) = \sin x \) from \( x = 0 \) to \( x = \pi/2 \) using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

Evaluate the integral by making the given substitution. \( \displaystyle \int \frac{x^3}{x^4 - 5} \, dx \), \( u = x^4 - 5 \)

Find the general indefinite integral. \( \displaystyle \int (x^{1.3} + 7x^{2.5}) \, dx \)

Sketch the area represented by \( g(x) \). Then find \( g'(x) \) in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. \( \displaystyle g(x) = \int^x_1 t^2\,dt \)

(a) Estimate the area under the graph of \( f(x) = 1 + x^2 \) from \( x = -1 \) to \( x = 2 \) using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left endpoints. (c) Repeat part (a) using midpoints. (d) From your sketches in parts (a)-(c), which appears to be the best estimate?

Evaluate the integral by making the given substitution. \( \displaystyle \int \sqrt{2t + 1} \, dt \), \( u = 2t + 1 \)

Find the general indefinite integral. \( \displaystyle \int \sqrt[4]{x^5} \, dx \)

Sketch the area represented by \( g(x) \). Then find \( g'(x) \) in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. \( \displaystyle g(x) = \int^x_0 (2 + \sin t)\,dt \)

(a) Graph the function $$ f(x) = x - 2 \ln x \hspace{10mm} 1 \le x \le 5 $$ (b) Estimate the area under the graph of \( f \) using four approximating rectangles and taking the sample points to be (i) right endpoints and (ii) midpoints. In each case sketch the curve and the rectangles. (c) Improve your estimates in part (b) by using eight rectangles.

Evaluate the indefinite integral. \( \displaystyle \int x \sqrt{1 - x^2} \, dx \)

Find the general indefinite integral. \( \displaystyle \int (5 + \frac{2}{3}x^2 + \frac{3}{4}x^3) \, dx \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle g(x) = \int^x_0 \sqrt{t + t^3} \,dt \)

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

Find the general indefinite integral. \( \displaystyle \int (u^6 - 2u^5 - u^3 + \frac{2}{7}) \, du \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle g(x) = \int^x_1 \ln (1 + t^2) \,dt \)

Evaluate the upper and lower sums for \( f(x) = 1 + x^2 \), \( -1 \le x \le 1 \), with \( n \) = 3 and 4. Illustrate with diagrams like Figure 14.

Evaluate the indefinite integral. \( \displaystyle \int (1 - 2x)^9 \, dx \)

Find the general indefinite integral. \( \displaystyle \int (u + 4)(2u + 1) \, du \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle g(s) = \int^s_5 (t - t^2)^8 \,dt \)

Use the Midpoint Rule with the given value of \( n \) to approximate the integral. Round the answer to four decimal places. \( \displaystyle \int^8_0 \sin \sqrt{x}\, dx \), \( n = 4 \)

With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of \( n \), using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for \( n \) = 10, 30, 50, and 100. Then guess the value of the exact area. The region under \( y = x^4 \) from 0 to 1

Evaluate the indefinite integral. \( \displaystyle \int \sin t \sqrt{1 + \cos t} \, dt \)

Find the general indefinite integral. \( \displaystyle \int \sqrt{t} (t^2 + 3t + 2) \, dt \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle h(u) = \int^u_0 \frac{\sqrt{t}}{t + 1} \,dt \)

Use the Midpoint Rule with the given value of \( n \) to approximate the integral. Round the answer to four decimal places. \( \displaystyle \int^1_0 \sqrt{x^3 + 1}\, dx \), \( n = 5 \)

With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of \( n \), using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for \( n \) = 10, 30, 50, and 100. Then guess the value of the exact area. The region under \( y = \cos x \) from 0 to \( \pi/2 \)

Evaluate the indefinite integral. \( \displaystyle \int \cos (\pi t/2) \, dt \)

Find the general indefinite integral. \( \displaystyle \int \frac{1 + \sqrt{x} + x}{x} \, dx \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle F(x) = \int^0_x \sqrt{1 + \sec t} \,dt \) $$ \biggl[ \textit{Hint:} \int^0_x \sqrt{1 + \sec t} \,dt = - \int^x_0 \sqrt{1 + \sec t} \,dt \biggr] $$

Use the Midpoint Rule with the given value of \( n \) to approximate the integral. Round the answer to four decimal places. \( \displaystyle \int^2_0 \frac{x}{x + 1}\, dx \), \( n = 5 \)

Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if \( x_{i}^{*} \) is a left or right endpoint. (For instance, in Maple use leftbox, rightbox, leftsum, and rightsum.) (a) If \( f(x) = 1/(x^2 + 1) \), \( 0 \le x \le 1 \), find the left and right sums for \( n \) = 10, 30, and 50. (b) Illustrate by graphing the rectangles in part (a). (c) Show that the exact area under \( f \) lies between 0.780 and 0.791.

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

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

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle R(y) = \int^2_y t^3 \sin t \,dt \)

Use the Midpoint Rule with the given value of \( n \) to approximate the integral. Round the answer to four decimal places. \( \displaystyle \int^{\pi}_0 x \sin^2x\, dx \), \( n = 4 \)

Evaluate the indefinite integral. \( \displaystyle \int \frac{dx}{5 - 3x} \)

Find the general indefinite integral. \( \displaystyle \int (\sin x + \sinh x)\, dx \)

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle h(x) = \int^{e^x}_1 \ln t \,dt \)

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.