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13. Evaluate: (emcos x dx. Hint: Notice we see sin x and its derivative cosx. u=sin...
Tutorial Exercise Evaluate the integral using the substitution rule. sin(x) 1/3 1* dx cos(x) Step 1...
Tutorial Exercise Evaluate the integral using the substitution rule. sin(x) 1/3 1* dx cos(x) Step 1 of 4 To integrate using substitution, choose u to be some function in the integrand whose derivative (or some constant multiple of whose derivative) is a factor of the integrand. Rewriting a quotient as a product can help to identify u and its derivative. 70/3 1." sin(x) dx = L" (cos(x) since) dx cos?(X) Notice that do (cos(x)) = and this derivative is a...
EXAMPLE 2 Find sin$(7x) cos”7x) dx. SOLUTION We could convert cos?(7x) to 1 - sin?(7x), but...
EXAMPLE 2 Find sin$(7x) cos”7x) dx. SOLUTION We could convert cos?(7x) to 1 - sin?(7x), but we would be left with an expression in terms of sin(7x) with no extra cos(7x) factor. Instead, we separate a single sine factor and rewrite the remaining sin" (7x) factor in terms of cos(7x): sin'(7x) cos”(7x) = (sinº(7x))2 cos(7x) sin(7x) = (1 - Cos?(7x))2 cos?(7x) sin(7x). in (7x) cos?(7x) and ich is which? Substituting u = cos(7x), we have du = -sin (3x) X...
1. a) Substitute u = sin(x) to evaluate sin^2(x) cos^3(x) dx. [trig identity sin2(x)+cos2(x) = 1]....
1. a) Substitute u = sin(x) to evaluate sin^2(x) cos^3(x) dx. [trig identity sin2(x)+cos2(x) = 1]. b) Find the antiderivatives: i) sin(2x) dx ii) (cos(4x)+3x^2) dx
13. Evaluate, S sin 5x cos x dx. Also prove that, 52" sin mx cos nx...
13. Evaluate, S sin 5x cos x dx. Also prove that, 52" sin mx cos nx dx = 0 Using reduction formula *****
Using reduction formula ***** 13. Evaluate, S sin 5x cos x dx. Also prove that, 53.4...
Using reduction formula *****
13. Evaluate, S sin 5x cos x dx. Also prove that, 53.4 sin mx cos nx dx = 0
Find the following trigonometric limit: lim sin - Hint the substitution [((u-1)E)s01 u= t-n makes life...
Find the following trigonometric limit: lim sin - Hint the substitution [((u-1)E)s01 u= t-n makes life 1. tm easier. Work inside the [...] first and then take the sine of your result, that is, use the rule that allows you to take the limit inside the sine function: lim sin(f(x)) n(Hm/s) sin x-a 2. Use the results we derived in class for power functions to find the derivative of g(x) (3x4 + v 4 atx Ans 3. When a function...
Evaluate the integral Z π 0 Z π x cos(y) y dy dx. Hint: Since cos(y)...
Evaluate the integral Z π 0 Z π x cos(y) y dy dx. Hint: Since cos(y) y doesn’t have an elementary antiderivative in y, the integral can only be evaluated by reversing the order of integration using Fubini’s theorem.
ENG 1005 ASSIGNMENTI QUESTIONS (1) Use integration by parts to calculate sin(In(x) dx and Here, In...
ENG 1005 ASSIGNMENTI QUESTIONS (1) Use integration by parts to calculate sin(In(x) dx and Here, In is the natural logarithm. cos(In(x))dx. [5 marks (2) (a) Use integration by parts on sinh(t) sinh(t)dt and the identity cosh (1) = 1+sinh'in to calculate the integral of sinh(r). (b) Calculate the integral of sinh(r) by expanding the product and then integrating, Confirm that you get the same answer as in part (a). (e) Show that if x is a positive real number, then...
In class we discussed the relationship between the hyperbolic functions and a hyperbola then showed that it is analogou...
In class we discussed the relationship between the hyperbolic functions and a hyperbola then showed that it is analogous to that of the trigonometric functions and a circle a. Derive an analogue to the Pythagorean Identities (cos2 x + sin2 x 1, etc. ) for the hyperbolic functions hint: Which hyperbola and which circle? (this will give you the relationship between cosh x and sinh x and the others are then easily found as they were in the case of...
Q1 2016 a) We want to develop a method for calculating the function f(x) = sin(t)/t...
Q1 2016
a) We want to develop a method for calculating the function f(x)
= sin(t)/t
dt
for small or moderately small values of x. this is a special
function called the sine integral, and it is related to another
special function called the exponential integral. it rises in
diffraction problems.
Derive a Taylor-series expression for f(x), and give an upper
bound for the error when the series is terminated after the n-th
order term. sint = see image
b)we...
Tutorial Exercise Evaluate the integral using the substitution rule. sin(x) 1/3 1* dx cos(x) Step 1 of 4 To integrate using substitution, choose u to be some function in the integrand whose derivative (or some constant multiple of whose derivative) is a factor of the integrand. Rewriting a quotient as a product can help to identify u and its derivative. 70/3 1." sin(x) dx = L" (cos(x) since) dx cos?(X) Notice that do (cos(x)) = and this derivative is a...
EXAMPLE 2 Find sin$(7x) cos”7x) dx. SOLUTION We could convert cos?(7x) to 1 - sin?(7x), but we would be left with an expression in terms of sin(7x) with no extra cos(7x) factor. Instead, we separate a single sine factor and rewrite the remaining sin" (7x) factor in terms of cos(7x): sin'(7x) cos”(7x) = (sinº(7x))2 cos(7x) sin(7x) = (1 - Cos?(7x))2 cos?(7x) sin(7x). in (7x) cos?(7x) and ich is which? Substituting u = cos(7x), we have du = -sin (3x) X...
13. Evaluate, S sin 5x cos x dx. Also prove that, 52" sin mx cos nx dx = 0 Using reduction formula *****
Using reduction formula *****
13. Evaluate, S sin 5x cos x dx. Also prove that, 53.4 sin mx cos nx dx = 0
Find the following trigonometric limit: lim sin - Hint the substitution [((u-1)E)s01 u= t-n makes life 1. tm easier. Work inside the [...] first and then take the sine of your result, that is, use the rule that allows you to take the limit inside the sine function: lim sin(f(x)) n(Hm/s) sin x-a 2. Use the results we derived in class for power functions to find the derivative of g(x) (3x4 + v 4 atx Ans 3. When a function...
ENG 1005 ASSIGNMENTI QUESTIONS (1) Use integration by parts to calculate sin(In(x) dx and Here, In is the natural logarithm. cos(In(x))dx. [5 marks (2) (a) Use integration by parts on sinh(t) sinh(t)dt and the identity cosh (1) = 1+sinh'in to calculate the integral of sinh(r). (b) Calculate the integral of sinh(r) by expanding the product and then integrating, Confirm that you get the same answer as in part (a). (e) Show that if x is a positive real number, then...
In class we discussed the relationship between the hyperbolic functions and a hyperbola then showed that it is analogous to that of the trigonometric functions and a circle a. Derive an analogue to the Pythagorean Identities (cos2 x + sin2 x 1, etc. ) for the hyperbolic functions hint: Which hyperbola and which circle? (this will give you the relationship between cosh x and sinh x and the others are then easily found as they were in the case of...
Q1 2016
a) We want to develop a method for calculating the function f(x)
= sin(t)/t
dt
for small or moderately small values of x. this is a special
function called the sine integral, and it is related to another
special function called the exponential integral. it rises in
diffraction problems.
Derive a Taylor-series expression for f(x), and give an upper
bound for the error when the series is terminated after the n-th
order term. sint = see image
b)we...
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