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- Solbe non homogeneous 2nd order DE using undefined coefficienta. Explain why 0 ≤ x²arctan(x) ≤ (pi*x²)/4 for all 0 ≤ x ≤ 1. b. Use the properties of the integrals to show that the value of the integral lower bound is 0, higher bound is 1 and the integral is x² arctan(x) dx lies on the interval [0,pi/12]a) Use the reduction formula twice to find the value of this integral: integral tan^(5)xdx Reduction formula:integral tan^(m)xdx=(tan^(m-1)x)/m-1 - integral tan^(m-2)xdx. b) Use the Product to Sum Identities and integration to find the Area of the Q1 region trapped between the graph of the function f(x) and the x – axis over the given interval. Sketch the region first and estimate its’ size before integrating in each case. 1) f(x)=4sin(3x)cosx, I: [0, pi/3] 2) f(x)= 6cos(2x)cosx, I: [0, pi/4]
- 6. Evaluate the definite integral. Use “u” substitution if applicableGeneral construction of Riemann integral of function f in ℝ³Using the residue theorem, evaluate the integral Show the details of your calculation clearly. Provide a sketch of your contour of choice (including the relevant singular point(s)) and check that the conditions for Jordan’s lemma to hold are satisfied before invoking the lemma.
- 1) again consider the definite integral 2) consider the integral use either of the two methods of substitutionScenario The work done by a sliding piston is given by the expression wd = ∫3t2 + 2t dt use integration to determine the work done by the piston between t = 1s and t = 3s. Given that the velocity of a body is given by v = u + at where u is the initial velocity, t is the time in seconds and a is the acceleration, use integration to derive an expression for the distance the body has travelled in t seconds. find ò cos(3x + 4) dx ( Hint Using the substitution u = 3x + 4. ) find dx ( Hint Using integration by parts )Python: Simpson's rule says the integral from x_0 to x_2 of f(x)dx is approximately h(1/3 f(x_0) + 4/3 f(x_1) _1/3 f(x_2)) where h = x_2-x_0 and x_1 is the midpoint of x_0 and x_2. Write a function simp(f,a,b,n) which integrates the function f(x) over the interval [a,b] by dividing it into n subintervals. integrate e^-x over [0,1] to make sure it matches the integral found with scipy.integrate.quad to 5 decimal places