Posted by weloo_volley on Wednesday, April 25, 2012 at 12:16am.heeeeelp meeeee ... (a)find antiderivatives for the following functions : (i) e^5x sinh3x + 4x+6/x^2+3x+5 . (ii) �ã3x(x^2-2/x+1). (b)Evaluate the following integrals over the given intervals: (i)4/2x-1 - 3/x+4 over [1,3] (ii)cosh 3x-sinh4x over [0, ln2] (c)A particle is moving from rest with an acceleration a(t)=3�ãt+1 m/sec^2 (i)Find the velocity of the particle after 25 seconds . (ii)Find the total distance travelled by the particle in the first 16 seconds.
(a) To find the antiderivatives for the given functions, you can use the rules of integration. Here are the solutions for each function:
(i) For the function e^5x sinh3x + 4x+6/x^2+3x+5:
- The antiderivative of e^5x is (1/5)e^5x.
- The antiderivative of sinh3x is (1/3)cosh3x.
- The antiderivative of 4x+6/x^2+3x+5 can be found using partial fraction decomposition.
- Once you have all the antiderivatives, you can combine them to find the overall antiderivative.
(ii) For the function √3x(x^2-2/x+1):
- The antiderivative of √3x is (2/3)√3(x^(3/2)).
- The antiderivative of (x^2-2/x+1) can be found using polynomial division.
- Once you have all the antiderivatives, you can combine them to find the overall antiderivative.
(b) To evaluate the integrals over the given intervals, you need to find the antiderivative of each function first. Then, you can use the fundamental theorem of calculus to calculate the definite integral over the given interval. Here are the solutions for each integral:
(i) For the function 4/(2x-1) - 3/(x+4) over [1, 3]:
- Find the antiderivative of each term: 4ln|2x-1| - 3ln|x+4|.
- Substitute the upper and lower limits into the antiderivative expression and subtract the results.
(ii) For the function cosh 3x - sinh 4x over [0, ln 2]:
- Find the antiderivative of each term: (1/3)sinh3x - (1/4)cosh4x.
- Substitute the upper and lower limits into the antiderivative expression and subtract the results.
(c) To solve the particle motion problem:
(i) Find the velocity of the particle after 25 seconds:
- Acceleration is the derivative of velocity. So, to find the velocity, integrate the acceleration function.
- Integrate 3√t + 1 with respect to t to get the velocity function.
- Substitute t = 25 seconds into the velocity function to find the velocity at that time.
(ii) Find the total distance traveled by the particle in the first 16 seconds:
- The total distance traveled can be obtained by integrating the absolute value of the velocity function over the given time interval.
- Integrate |3√t + 1| with respect to t over the interval [0, 16] to find the total distance traveled.