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(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 antiderivatives for the given functions, we can use the rules of integration.
(i) Start by finding the antiderivative of each term separately:
- The antiderivative of e^5x is (1/5)e^5x.
- The antiderivative of sinh(3x) is (1/3)cosh(3x).
- The antiderivative of 4x is 2x^2.
- For the rational function (6/x^2+3x+5), you can use partial fraction decomposition to separate it into simpler fractions: A/x + B/(x^2+3x+5). After finding the values of A and B, you can find the antiderivative.
Then, you can add all the individual antiderivatives together to get the antiderivative of the entire function.
(ii) The antiderivative of 3x(x^2-2)/(x+1) is a bit more complicated, but it can be found using techniques such as u-substitution or integration by parts.
(b) To evaluate the given integrals over the given intervals, we can use the definite integral formula and apply the Fundamental Theorem of Calculus.
(i) Start by finding the antiderivative of each term separately, just like in part (a). Then, substitute the upper and lower limits of the integral into the antiderivative expression and subtract the result for the lower limit from the result for the upper limit.
(ii) Follow the same steps as in part (i), but with the appropriate antiderivative expression for the given function.
(c) To solve the problems related to particle motion, we can use the basic equations of motion in calculus.
(i) The velocity of a particle is the antiderivative of its acceleration function. So, you need to find the antiderivative of a(t) = 3√t + 1. Once you find it, substitute t = 25 into the antiderivative expression to get the velocity of the particle after 25 seconds.
(ii) The total distance travelled by the particle is found by integrating the absolute value of its velocity function over the given time interval. In this case, integrate the absolute value of the velocity function from t = 0 to t = 16 to find the total distance travelled by the particle in the first 16 seconds.