Let I = dx x2 4x + 8 . (a) Complete the square to show that x2 4x + 8 = (x 2)2 + 4. (b)

State the trigonometric substitution appropriate to the given integral: (a) 9 x2 dx (b) x2(x2 16) 3/2 dx (c) x2(x2 + 16) 3/2 dx (d) (x2 5) 2 dx 2.

Is trigonometric substitution needed to evaluate x 9 x2 dx?

Express sin 2 in terms of x = sin .

Draw a triangle that would be used together with the substitution x = 3 sec .

In Exercises 510, use the indicated substitution to evaluate the integral. 5. 16 5x2 dx, x = 4 5 sin

In Exercises 510, use the indicated substitution to evaluate the integral.1/2 0 x2 1 x2 dx, x = sin 7

In Exercises 510, use the indicated substitution to evaluate the integral.dx x x2 9 , x = 3 sec

In Exercises 510, use the indicated substitution to evaluate the integral.1 1/2 dx x2 x2 + 4 , x = 2 tan

In Exercises 510, use the indicated substitution to evaluate the integral.dx (x2 4)3/2 , x = 2 sec

In Exercises 510, use the indicated substitution to evaluate the integral.1 0 dx (4 + 9x2)2 , x = 2 3 tan

Evaluate xdx x2 4 in two ways: using the direct substitution u = x2 4 and by trigonometric substitution.

Is the substitution u = x2 4 effective for evaluating the integral x2 dx x2 4 ? If not, evaluate using trigonometric substitution.

Evaluate using the substitution u = 1 x2 or trigonometric substitution. (a) x 1 x2 dx (b) x2 1 x2 dx (c) x3 1 x2 dx (d) x4 1 x2 dx 14.

Evaluate: (a) 1 0 dt (t2 + 1)3/2 (b) 1 0 t dt (t2 + 1)3/2 I

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. 15. x2 dx 9 x2

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dt (16 t2)3/2

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dx x x2 + 16

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.12 + 4t2 dt

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dx x2 9

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dt t2 t2 25

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dy y2 5 y2

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.x3 9 x2 dx

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dx 25x2 + 2

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dt (9t2 + 4)2

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dz z3 z2 4

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dy y2 9

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.x2 dx (6x2 49)1/2 2

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.dx (x2 4)2

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.1 0 dt (t2 + 9)2 3

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.1 0 dx (x2 + 1)3

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.x2 dx (x2 1)3/2 3

In Exercises 1532, evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary.x2 dx (x2 + 1)3/2

Prove for a > 0: dx (x2 + a)2 = 1 2a x x2 + a + 1 a tan1 x a + C

Prove for a > 0: dx (x2 + a)2 = 1 2a x x2 + a + 1 a tan1 x a + C

Let I = dx x2 4x + 8 . (a) Complete the square to show that x2 4x + 8 = (x 2)2 + 4. (b) Use the substitution u = x 2 to show that I = du u2 + 22 . Evaluate the u-integral. (c) Show that I = ln  (x 2)2 + 4 + x 2 + C.

Evaluate dx 12x x2 . First complete the square to write 12x x2 = 36 (x 6) 2.

In Exercises 3742, evaluate the integral by completing the square and using trigonometric substitution. 37. dx x2 + 4x + 13

In Exercises 3742, evaluate the integral by completing the square and using trigonometric substitution.dx 2 + x x2

In Exercises 3742, evaluate the integral by completing the square and using trigonometric substitution.dx x + 6x2

In Exercises 3742, evaluate the integral by completing the square and using trigonometric substitution.x2 4x + 7 dx 4

In Exercises 3742, evaluate the integral by completing the square and using trigonometric substitution.x2 4x + 3 dx 4

In Exercises 3742, evaluate the integral by completing the square and using trigonometric substitution.dx (x2 + 6x + 6)2

In Exercises 4346, evaluate using Integration by Parts as a first step. 43. sec1 xdx 4

In Exercises 4346, evaluate using Integration by Parts as a first step.sin1 x x2 dx 4

In Exercises 4346, evaluate using Integration by Parts as a first step.ln(x2 + 1)dx 4

In Exercises 4346, evaluate using Integration by Parts as a first step.x2 ln(x2 + 1)dx

Find the average height of a point on the semicircle y = 1 x2 for 1 x 1.

Find the volume of the solid obtained by revolving the graph of y = x 1 x2 over [0, 1] about the y-axis.

Find the volume of the solid obtained by revolving the region between the graph of y2 x2 = 1 and the line y = 2 about the line y = 2.

Find the volume of revolution for the region in Exercise 49, but revolve around y = 3.

Compute dx x2 1 in two ways and verify that the answers agree: first via trigonometric substitution and then using the identity

You want to divide an 18-in. pizza equally among three friends using vertical slices at x as in Figure 7. Find an equation satisfied by x and find the approximate value of x using a computer algebra system.

A charged wire creates an electric field at a point P located at a distance D from the wire (Figure 8). The component E of the field perpendicular to the wire (in newtons per coulomb) is E = x2 x1 kD (x2 + D2)3/2 dx where is the charge density (coulombs per meter), k = 8.99 109 Nm2/C2 (Coulomb constant), and x1, x2 are as in the figure. Suppose that = 6 104 C/m, and D = 3 m. Find E if (a) x1 = 0 and x2 = 30 m, and (b) x1 = 15 m and x2 = 15 m. x1 x2 P D y x

Let Jn = dx (x2 + 1)n . Use Integration by Parts to prove Jn+1 = 1 1 2n Jn + 1 2n x (x2 + 1)n

Prove the formula 1 x2 dx = 1 2 sin1 x + 1 2 x 1 x2 + C