Question: Evaluate the integral by completing the square

> Find the area of the region bounded by the hyperbola 9x2 - 4y2 = 36 and the line x = 3.

> Evaluate (a) by trigonometric substitution. (b) by the hyperbolic substitution x = a sinh t. x2 (x² + a²)½ dx

> (a) Use trigonometric substitution to show that (b) Use the hyperbolic substitution x = a sinh t to show that These formulas are connected by Formula 3.11.3. dx = In(x + Vx? + a²) + C x² + a² -2 (). dx sinh + C x² + a² a

> Evaluate the integral. cos t = dt V1 + sin?t */2 Jo

> For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer)? (a) 2 (b) E 2" (-3)*-1 (c) E Vn (d) E n-1 1 + n? n-1 2 - |

> Evaluate the integral. JxVī - x* dx

> Evaluate the integral. x? + 1 (x² – 2x + 2)

> Evaluate the integral. S | Vx? + 2x dx

> Evaluate the integral. - dx J (3 + 4x – 4x²)³/2

> Evaluate the integral. x² V3 + 2x – x² dx

> Evaluate the integral. x² dx

> Evaluate the integral. dx Vx? + 2x + 5

> Evaluate the integral. I Vx? + 1 dx

> Evaluate the integral. x2 r0.6 dx V9 - 25х?

> Evaluate the integral. dx V1 + x?

> Evaluate the integral. V1 + x? dx

> Evaluate the integral. dx I [(ax)? – b²]³/2

> Evaluate the integral. dx (2/3 JAs x/9x? – 1 J2/3

> Evaluate the integral. r Va² – x² dx

> Evaluate the integral. dx (x² + 1)?

> Evaluate the integral. Vx? – 9 dx 2 .3

> Evaluate the integral. dt V4 + t?

> Evaluate the integral. r1/2 x V1 – 4x² dx

> Evaluate the integral. 2/3 /4 - 9х? dx Jo

> Evaluate the integral. dx (x² 3/2

> Evaluate the integral. dt t?V12 – 16

> Evaluate the integral. dx Jo Ta² + x²)/2* a >0

> Evaluate the integral. dx (36— х2

> Evaluate the integral. .2 dx .4

> Evaluate the integral. x2 dx

> Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. /x² – 4 dx X = 2 sec 0

> Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. x x = 2 tan 0 dx Vx2 + 4

> Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. dx x = 2 sin 0 J x4 - x2

> If a ≠ 0 and n is a positive integer, find the partial fraction decomposition of S(x) x"(х — а)

> If f is a quadratic function such that f(0) = 1 and is a rational function, find the value of f(0). f(x) ·dx x²(x + 1)'

> Suppose that F, G, and Q are polynomials and for all x except when Q(x) = 0. Prove that F(x) = G(x) for all x. F(x) G(x) Q(x) Q(x)

> (a) Use integration by parts to show that, for any positive integer n (b) Use part (a) to evaluate dx dx J (x² + a²)* 2a*(n – 1)(x² + a²)" -| 2n – 3 dx 2a*(n – 1) J 2а"(л — (x² + a²)ª ! dx dx and (x² + 1)² (x² + 1)'

> (a) Find the partial fraction decomposition of the function (b) Use part (a) to find / and graph f and its indefinite integral on the same screen. (c) Use the graph of f to discover the main features of the graph of / 12x – 7x – 13x² + 8 100x – 80x +

> (a) Use a computer algebra system to find the partial fraction decomposition of the function (b) Use part (a) to find / (by hand) and compare with the result of using the CAS to integrate f directly. Comment on any discrepancy. 4x – 27x² + 5x – 32 f(

> Factor x4 + 1 as a difference of squares by first adding and subtracting the same quantity. Use this factorization to evaluate /

> One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring. (The photo shows a screw-worm fly, the first pest eff

> Find the volume of the resulting solid if the region under the curve / is rotated about (a) the x-axis and (b) the y-axis.

> Find the area of the region under the given curve from 1 to 2. х2 + 1 y 3x – x?

> Find the area of the region under the given curve from 1 to 2. 1 y = x' + x

> Use the substitution to transform the integrand into a rational function of t and then evaluate the integral. 12 sin 2x dx w/2 Jo 2 + cos x

> Use the substitution to transform the integrand into a rational function of t and then evaluate the integral. /2 dx /3 1 + sin x – cos x

> Use the substitution to transform the integrand into a rational function of t and then evaluate the integral. 1 dx 3 sin x – 4 cos x

> Use the substitution to transform the integrand into a rational function of t and then evaluate the integral. dx 1 - cos x

> The German mathematician Karl Weierstrass (1815–1897) noticed that the substitution t = tan(x/2) will convert any rational function of sin x and cos x into an ordinary rational function of t. (a) If / sketch a right triangle or use trig

> Evaluate the integral by completing the square and using Formula 6. dx x² – 2x

> Use integration by parts, together with the techniques of this section, to evaluate the integral. x tan¯'x dx

> (a) Show that the function is a solution of the differential equation (b) Show that f(x) = ex. S(x) = E o n! f'(x) = f(x)

> Use integration by parts, together with the techniques of this section, to evaluate the integral. | In(x? – x + 2) dx In(x²

> Make a substitution to express the integrand as a rational function and then evaluate the integral. cosh t dt J sinh?t + sinh't

> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx 1+ e*

> Make a substitution to express the integrand as a rational function and then evaluate the integral. e* (e* – 2)(e²* + 1) ,2x

> Make a substitution to express the integrand as a rational function and then evaluate the integral. sec?t -dt tan?t + 3 tan t + 2

> Make a substitution to express the integrand as a rational function and then evaluate the integral. sin x dx I cos'x – 3 cos x

> Make a substitution to express the integrand as a rational function and then evaluate the integral. ,2x e2х + Зе* + 2

> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx

> Make a substitution to express the integrand as a rational function and then evaluate the integral. 1 - dx [Hint: Substitute u =

> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx (1 + JF)²

> Show that the function is a solution of the differential equation S(x) = § (-1)*x 2n Σ (2n)! (-1)"x²" f"(x) + f(x) = 0

> (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor’s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (c) Check your result in part (b) by graphing / S

> (a) Find the Taylor polynomials up to degree 3 for f(x) = tan x centered at a = 0. Graph f and these polynomials on a common screen. (b) Evaluate f and these polynomials at / and / (c) Comment on how the Taylor polynomials converge to f(x).

> Make a substitution to express the integrand as a rational function and then evaluate the integral. .3 Vx? dx + 1

> Make a substitution to express the integrand as a rational function and then evaluate the integral. 1 dx Jo 1+ x

> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx x² + xF 2.

> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx 2 Vx + 3 + x

> Make a substitution to express the integrand as a rational function and then evaluate the integral. dx x/x - 1

> Evaluate the integral. ' + 2x² + 3x – 2 dx (x² + 2x + 2)²

> Evaluate the integral. x? — Зх + 7 (x? — 4х + 6)?

> Evaluate the integral. x* + 3x? + 1 x* + 5x + 5x

> Evaluate the integral. 5x + 7x? + x + 2 dx x(x² + 1)²

> Evaluate the integral. x* + x – 1 x³ + 1

> Use the result of Example 7 to compute arctan 0.2 correct to five decimal places.

> Evaluate the integral. x' + 2x x* + 4x² + 3

> Evaluate the integral. dx x2 + 4x + 13 Jo

> Evaluate the integral. dx x³ – 1 3

> Evaluate the integral. x - 2x? + 2x – 5 dx x* + 4x² + 3

> Evaluate the integral. x + 4 dx J x? + 2x + 5

> Evaluate the integral. x3 + 6х — 2 dx x* + 6x²

> Evaluate the integral. x' + 4x + 3 dx x* + 5x? + 4

> Evaluate the integral. x? + x + 1 dx (x² + 1)²

> Evaluate the integral. 4x dx x² + x + 1 .3 x' +

> Evaluate the integral. x? - x + 6 dx x' + 3x

> Use a power series to approximate the definite integral to six decimal places. ro.3 dx 1+ x*

> Evaluate the integral. 10 dx (x – 1)(x² + 9) .2

> Evaluate the integral. x* + 9x? + x + 2 dx x² + 9

> Evaluate the integral. dt (1? – 1)?

> Evaluate the integral. x(3 — 5х) dx J (3x – 1)(x – 1)²

> Evaluate the integral. x? 1 + x + dx Jo (x + 1)°(x + 2)

> Evaluate the integral. Зx2 + 6х + 2 dx х? + 3х + 2

> Evaluate the integral. 4y? — Ту — 12 dy Л у(у + 2)(у — 3)

> Evaluate the integral. x' + 4x? + x - 1 dx x' + x?

> Evaluate the integral. x - 4x + 1 · dx 1 x² – 3x + 2

> Evaluate the integral. 1 (x + a)(x + b)

> Use a power series to approximate the definite integral to six decimal places. c0.2 x In(1 + x²) dx

> Evaluate the integral. x - 4 dx -o?? - 5x + 6

> Evaluate the integral. 2 dx Jo 2x + 3x + 1