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mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Evaluate ∫(sin−1 x)2 dx. Use Integration by Parts first and then substitution.
Evaluate ∫x7 cos(x4) dx.
Evaluate ∫10 x3e−x2 dx.
Find the area of the region that lies under the graph of y = (5 − x) ln x and above the x-axis.
The present value (PV) of an investment that provides income continuously at a rate R(t) $/year for T years, and earns interest at rate r, isWe think of present value as the payment that we would need to receive at t = 0 so that at time T the payment’s value would be the same as the amount
Find the area enclosed by y = ln x and y = (ln x)2.
Derive the reduction formula (In x) dx = x(lnx)* -k (lnx)k-1 dx
Derive the reduction formulas Sx fx cos xdx = x Sx nf x x" cos x dx = x" sin x - n x−1 sin xdx x" sin x dx = -x" cos x + n +nfx²1 Sx x-1 cos x dx
Prove that ∫xbx dx = bx 1 nby²). X Inb (In b)2, + C.
The Integration by Parts formula can be written Suvd -Sve uv dx = UV - uV- V du
Define Pn(x) byUse the reduction formula in Exercise 64 to prove that Pn(x) = xn − nPn−1(x). Use this recursion relation to find Pn(x) for n = 1, 2, 3, 4. P0(x) = 1.Data From Exercise 64 [x¹e² dx = P₁ xe* dx= Pn(x) et + C
Assume that ƒ(0) = ƒ(1) = 0 and that ƒ"exists. ProveUse this to prove that if ƒ(0) = ƒ(1) = 0 and ƒ"(x) = λ ƒ(x) for some constant λ, then λ [² f(x)f(x) dx = -f² f'(x)7² dx
Set I(a, b) = ∫10xa(1 − x)b dx, where a, b are whole numbers. (a) Use substitution to show that I(a, b) = 1(b, a). 1 (b) Show that I(a,0) = I(0, a) = a + 1 (c) Prove that for a ≥ 1 and b≥ 0, (e) Show that I(a, b) = I(a, b) = a b + 1 (d) Use (b) and (c) to calculate I(1, 1) and I(3, 2). a!
Let In = ∫xn cos(x2) dx and Jn = xn sin(x2) dx. (a) Find a reduction formula that expresses I, in terms of Jn-2. Write x" cos(x²) as x-1(x cos(x²)). (b) Use the result of (a) to show that I, can be evaluated explicitly if n is odd. (c) Evaluate 13.
Evaluate the integral by following the steps given. dx x² √x² - 2 I = = S₁
Evaluate the integral by following the steps given.(a) Show that the substitution x = 3 sin θ transforms I into ∫dθ, and evaluate I in terms of θ.(b) Evaluate I in terms of x. = S I = dx √9 - x²
Evaluate the integral by following the steps given. = S I = dx √4x² +9
Evaluate the integral by following the steps given. I = dx (x² + 4)² S
Use the indicated substitution to evaluate the integral. S √16-5x² dx, X = 4 √5 sin 0
Use the indicated substitution to evaluate the integral. 1/2 x² √1-x² 50¹/² dx, x = sin 0
Use the indicated substitution to evaluate the integral. dx x√√x²-9² x = 3 sec 0
Use the indicated substitution to evaluate the integral. dx S1/² √₁/2 x² √√√x² +4² x = 2 tan 0
Use the indicated substitution to evaluate the integral. dx (x²-4)3/2¹ Sa x = 2 sec 0
Use the indicated substitution to evaluate the integral. So 10 dx (4 + 4x²)²³ x = tan 0
Is the substitution u = x2 − 4 effective for evaluating the integral If not, evaluate using trigonometric substitution. s x dx √x²-4 -?
Evaluate ∫x dx/√x2 − 4 in two ways: using the direct substitution u = x2 − 4 and by trigonometric substitution.
Evaluate using the substitution u = 1 − x2 or trigonometric substitution. X (a) S √ ₁ - ² dx V1-x² (c) fx³¹ √T x³√1-x² dx (b) (d) fx²√₁-x²dx √1 - x² dx A S √1 x² dx
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. S x² dx √9 - x²
Evaluate: (a) ∫10 dt/(t2 + 1)3/2 (b) ∫10 t dt/(t2 + 1)3/2
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dt (16-12)³/2 S
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dx x√x² xVx² + 16
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. S V12 + 4t² dt
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dx √x²-9
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dt 1² √1²-25 S
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dy 1² √√√5-y² S
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. S2²³ √9 x³ √9 - x² dx
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dx √25x² + 2
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dt (9f² + 4)² S
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dz z³ √√z² - 4 S
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dy √y²-9 S
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. x² dx (6x² - 49)1/2 S
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. dx (x² - 4)² [
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. S 10 dt (1² + 9)²
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. S'; 0 dx (x² + 1)³
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. S x² dx (x² - 1)3/2
Evaluate using trigonometric substitution. Refer to the table of trigonometric integrals as necessary. / x² dx (x² + 1)³/2
(a) Using a trigonometric substitution, compute the integral and show that for a > 0(b) Verify the formula via differentiation. dx x² + a² S 1 - D tan - + C a
Compute ∫r0 1/(4 − x2)3/2 dx, expressing the result in terms of r. Then, discuss what happens to the value of the definite integral as r approaches 2.
(a) Using a trigonometricsubstitution, compute the integral and show that for a > 0(b) Verify the formula via differentiation. dx 1 1 S (8² + 6²)² = 26² ( x² + a² + tan²-¹ x) + C - a
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 Ι√(x − 2)2 + 4 + x − 2Ι + C.
Evaluate the integral by completing the square and using trigonometric substitution. S dx √x² + 4x + 13
Evaluate ∫ dx/√12x − x2 . First, complete the square to write 12x − x2 = 36 − (x − 6)2.
Evaluate the integral by completing the square and using trigonometric substitution. dx √2+x-x² S
Evaluate the integral by completing the square and using trigonometric substitution. dx S√√x + 6x²
Evaluate the integral by completing the square and using trigonometric substitution. √ √x² - 4x + 7 dx
Evaluate the integral by completing the square and using trigonometric substitution. S √x² - 4x + 3dx
Evaluate the integral by completing the square and using trigonometric substitution. C dx (x² + 6x + 6)²
Find the volume of the solid obtained by revolving the graph of y = x √1 − x2 over [0, 1] about the y-axis.
You want to divide an 18 inch pizza equally among three friends using vertical slices at ±x as in Figure 8. Find an equation satisfied by x and find the approximate value of x using a computer algebra system. -9 X 9 x
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.
A charged wire creates an electric field at a point P located at a distance D from the wire (Figure 9). The component E⊥ of the field perpendicular to the wire (in newtons per coulomb) iswhere λ is the charge density (coulombs per meter), k = 8.99 × 109 N · m2/C2 (Coulomb constant), and x1,
Find the volume of revolution for the region in Exercise 51, but revolve around y = 3.Data From Exercise 51Find 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.
Let Jn = ∫ dx/(x2 + 1)n.(a) Compute J1.(b) Use Integration by Parts to prove(c) Use this recursion relation to calculate J2 and J3. X Jan+1 = (1 - 2 ) x + (2n) (x² + 1)² Jn 2n
The area function F(x) = ∫x 0 √1 − t2 dt is an antiderivative of ƒ(x) = √1 − x2. Prove the formulausing geometry by interpreting the integral as the area of part of the unit circle. 1 1 *VI-Pdt = sin ¹² x + x V₁ -² √ Vi 2 S 0
Evaluate using methods similar to those that apply to the integrals of tanm x secn x. Sc cot³ x dx
Compute the area under the graph of y = cos3 x from x = 0 to x = π/2.
Evaluate using methods similar to those that apply to the integrals of tanm x secn x. fcs csc4 x dx
Compute the area under the graph of y = sin5 x from x = 0 to x = π.
Evaluate using methods similar to those that apply to the integrals of tanm x secn x. Scot³ x cot5 x csc² x dx
Evaluate using methods similar to those that apply to the integrals of tanm x secn x. 5 cot x csc x dx
Compute the area under the graph of y = tan2 x from x = 0 to x = π/4.
Compute the area under the graph of y = sec4 x from x = 0 to x = π/3.
Use the identities for sin 2x and cos 2x on page 415 to verify that the following formulas are equivalent: fsint xdx=(12) 32 fsint. sin x dx 1 -- 4 (12x 8 sin 2x + sin 4x) + C sin³ x cos x- nh 3 3 sin x cos x + = x + C x+³x+C 8
For n a positive integer, compute the area under the graph of y = sinn x cos3 x for 0 ≤ x ≤ π/2.
For n a positive integer, compute the area under the graph of y = tann x sec4 x for 0 ≤ x ≤ π/4.
Evaluate ∫ sin2 x cos3 x dx using the method described in the text and verify that your result is equivalent to the following result produced by a computer algebra system: fsin sin² x cos³ x dx = 1 30 (7 + 3 cos 2x) sin³ x + C
Use the substitution u = csc x − cot x to evaluate ∫ csc x dx (see Example 6). EXAMPLE 6 Integral of Secant Derive the formula [secx secx dx = In |secx + tan x + C
Find the volume of the solid obtained by revolving y = sin x for 0 ≤ x ≤ π about the x-axis.
Let Im = ∫π/20 sinm x dx.(a) Show that I0 = π/2 and I1 = 1.(b) Prove that, for m ≥ 2,(c) Use (a) and (b) to compute Im for m = 2, 3, 4, 5. Im = m-1 m -Im-2
Find the volume of the solid obtained by revolving y = tan x for 0 ≤ x ≤ π/6 about the x-axis.
This is a continuation of Exercise 76.Data From Exercise 76Set Im = ∫π/20 sinm x dx. Use Exercise 69 to prove thatConclude thatData From Exercise 69Let Im = ∫π/20 sinm x dx.Show that I0 = π/2 and I1 = 1.Prove that, for m ≥ 2, (a) Prove that I2m+1 ≤ 12m ≤
A 100-watt (W) light bulb has resistance R = 144 ohms (Ω) when attached to household current, where the voltage varies as V = V0 sin(2π ƒ t) (V0 = 110 V, ƒ = 60 Hz). The energy (in joules) expended by the bulb over a period of T seconds iswhere P = V2/R (J/s) is the power. Compute U if the bulb
Evaluate ∫π0 sin2 mx dx for m an arbitrary integer.
Let m, n be integers with m ≠ ±n. Use Eqs. (15)–(17) to prove the so-called orthogonality relations that play a basic role in the theory of Fourier Series (Figure 1): A A J y = sin 2x sin 4x J J T 21 sin mx sin nx dx = 0 cos mx cos nx dx = 0 sin mx cos nx dx = 0 W y = sin 3x cos 4x 21
Evaluate ∫ sin x ln(sin x) dx. Use Integration by Parts as a first step.
Set Im = ∫π/20 sinm x dx. Use Exercise 69 to prove thatConclude thatData From Exercise 69Let Im = ∫π/20 sinm x dx.(a) Show that I0 = π/2 and I1 = 1.(b) Prove that, for m ≥ 2,(c) Use (a) and (b) to compute Im for m = 2, 3, 4, 5. 12m 12m+1 = 2m 12m3 2m 2m-2 2m 2m-2 2m + 1 2m
Use Integration by Parts to prove that (for m ≠ 1) fsec sec" x dx: tan x secm-2 X m - 1 + m-2 m-1 se secm-2 x dx
Evaluate the integral using the Integration by Parts formula with the given choice of u and dv. | x sin x dx; u = x, dv = sin x dx
Evaluate the integral using the Integration by Parts formula with the given choice of u and dv. xe xe²x dx, u = x, dv = e²x dx
Evaluate the integral using the Integration by Parts formula with the given choice of u and dv. So (2x + 9)e* dx; u = 2x + 9, dv = e* dx
Evaluate the integral using the Integration by Parts formula with the given choice of u and dv. Sx x cos 4x dx; u = x, dv = cos 4x dx
Evaluate the integral using the Integration by Parts formula with the given choice of u and dv. Sx x³ In x dx; u = ln x, dv = x³ dx
Evaluate the integral using the Integration by Parts formula with the given choice of u and dv. Stan tan`1xdx; u = tan"1 x, dv = dx
Evaluate using substitution and then Integration by Parts. Seved J e√x dx Let u = x¹/2
Evaluate using substitution and then Integration by Parts. x³e¹² dx
Use Eq. (5) to find Eq.(5) S si sin³ x dx.
For ∫ x tan x dx, try Integration by Parts with u = x, dv = tan x dx and with u = tan x, dv = x dx, and describe the difficulty that you encounter in each case, keeping you from finding an antiderivative. There is no antiderivative formula for x tan x involving elementary functions.
For ∫ x sec x dx, try Integration by Parts with u = x, dv = sec x dx and with u = sec x, dv = x dx, and describe the difficulty that you encounter in each case, keeping you from finding an antiderivative. There is no antiderivative formula for x sec x involving elementary functions.
Derive the reduction formula I cos cos" x dx : = 1 - n n-1 +"=¹ fcos- n cos-xsin x + cos"-2 x dx
Use the reduction formula from Exercise 62 to find Data From Exercise 62Derive the reduction formula Sc cos³ x dx.
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