Express the integrand as a sum of partial fractions and...

Express the integrand as a sum of partial fractions and evaluate the integral. ? (3x^2 + x + 5) / ((x^2 + 3)(x - 4)) dx 3 ln |x - 4| + (?3 / 3) tan?¹(x?3 / 3) + C 3 ln |x - 4| + tan?¹(x?3 / 3) + C 3 ln |x - 4| + (1 / 3) tan?¹(x / 3) + C ln |x - 4| + (?3 / 3) tan?¹(x?3 / 3) + C QUESTION 11 Use the Trapezoidal Rule with n = 4 steps to estimate the integral. ? from -1 to 0 sin ?t dt (1 + ?2) / 2 - (1 + ?2) / 8 - (1 - ?2) - (1 + ?2) / 4

Transcript

00:01 Hello everyone, in the first part of the question we are given with integral of 3x square plus x plus 5 divided by x square plus 3 into x minus 4 dx.

00:16 So, let us solve this integral value.

00:19 Now, let us take 3x square plus x plus 5 divided by x square plus 3 into x minus 4 will be equal to we can consider this as ax plus b divided by x square plus 3 plus c by x minus 4.

00:39 So, this can be written as ax plus b into x minus 4 plus c into x square plus 3 divided by x square plus 3 into x minus 4.

00:56 So, here we will have 3x square plus x plus 5 divided by x square plus 3 into x minus 4.

01:06 So, as we see here the denominator are equal so that we can equate the numerator to find the value of a b c.

01:13 So, that 3x square plus x plus 5 will be equal to ax plus b into x minus 4 plus c into x square plus 3.

01:28 Now, equating this and solving we will get a is equal to 0 b is equal to 1 and c is equal to 3.

01:39 Now, substituting those values in the equation we will get ax plus b divided by x square plus 3 plus c divided by x minus 4 will become 1 divided by x square plus 3 plus 3 divided by x minus 4.

02:04 Now, let us take the integration.

02:09 So, integrating 1 divided by x square plus 3 with respect to dx and integrating 3 divided by x minus 4 dx will give 1 by root 3 tan inverse of root 3 by 3 x plus 3 log x minus 4 plus.

02:42 So, the correct option from the following we can choose the answer is 3 log x minus 4 plus root 3 by 3 tan inverse of x root 3 by 3 plus c which is the answer for the first part of our question.

03:19 In the second part of the question we have to use trapezoidal rule to find the integral value of the given function.

03:32 So, by trapezoidal rule we have the formula that will be equal to delta x by 2 into f of x naught plus 2 f of x 1 plus etcetera plus 2 f of x n minus 1 plus f of x n...

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