Answer to question 4.3.14

 

 Please review the section 4.3 on the book before going through the solution.

Let us put u = ln(tan(x)) and dv = sin(x)

whence 

and v = -cos(x)

Doing integration by parts:

Let I₂ = ∫1/sin(x)dx. This can be integrated as follows:

Make the substitution cos(x) = t:

Note that we use formula (10) of section 4.1 on the book to integrate 1/(u^2-a^2) form below.

 Using

We can simplify I₂ as:

Now we can get the result as I = -cos(x)ln(tan(x)) + I₂

Answer to question 5.1.14

Answer to 5.1.14

We have the question,

 

Since the fraction is improper, we need to single out the integral part(Refer 5.1.3 solved example in the textbook). Dividing numerator by denominator, we obtain :

Hence using this, our integral becomes :

Expand the remaining proper fraction into simple ones using partial fraction :

Hence :

 

Putting x = -1, we get 2A = -1 + 1 +2, or, A = 1

Now, expanding and comparing coefficients of similar terms on both sides and solving the equation, we get B = -2 and C = 1.

So,we have :

So, the integral becomes :

Integrating, we get the answer as :

Answer to question 2.2.8 all unsolved parts

Answer to 2.2.8 (e)

Differentiating this using quotient rule and chain rule, we get :

Now we need to use this result for simplifying the above differential :

From these two properties, sinh(bx)-cosh(bx) can be written as(This is obtained by simply subtracting) :

 

 So by replacing, our derivative is simplified as follows :  

NOTE : The answer given at the end of the book is wrong. The one above is correct.

Answer to 3.6.1 all unsolved subparts

Answer to 3.6.1 (b) :

Differentiating, we get :

Now we need to factorize this. By trail and error, we find that derivative of f(x) is zero when x=1 and hence (x-1) is a factor of it. Using long division by (x-1) we obtain :

Equating the derivative with zero, we get the critical points as follows :
x1=1, x2=2 and x3 = 3

Now I know you can do the rest ;)

Answer to 3.61 (d) :

To make the evaluation simple, let's subtract both sides by -1 and then the equation becomes :

 Differentiating, we get :

Equating it with zero, we get :
x=7/5
Now do the rest for yourself.

Answer to 4.2.14 all subparts

The answer given for some of the sub question in the book have some mistakes. I'm posting the corrected solution here.

Answer to 4.2.14 (a) :

The solution given in the book for this question is incorrect. You can solve the question as follows.

Substitute 1-x = t. Then dx=-dt . Now the substituted integrand should look like this :

Now, integrating term by term, we'll get :

Taking out the common term, we get :

Now replacing t with 1-x, you'll get the answer as :

This should be the answer. The answer given on the book has a mistake.

Answers to 4.2.13 all subparts

Since we have already discussed enough problems like this, I'll just give you hints on what to substitute and you'll have to do the rest(which is really simple) by yourself. Ask in the comments if you need any help by the way.

Answer to 4.2.13 (a) :

Substitute, 1 + lnx = t and then dx = x.dt. The integrand will now be as simple as t^(1/3)dt . Integrate it and replace t with 1 + lnx to obtain final answer.

Answer to 4.2.13 (b) :

Substitute, lnx = t then, dx = xdt . Now the integrand become dt/t . Integrate it and you'll get the answer.

Answer to 4.2.13 (c) :

Substitute x^2 = t then dx = dt/2x. Now the integral will look like this :

Now use the standard formula given below to integrate the substituted integrand :

 And then, replace t by x^2 to obtain final answer.

Answer to 4.2.14 (d) :

Use substitution given below : 

Now the integral becomes :

 Use the below standard integration formula to integrate the above integrand :

And after this, replace t with x^n to obtain the final answer.

Answer to 4.2.14 (e) :

Substitute squareroot(x) = t then, dx = 2t.dt
With the above substitutions, the question now has become really simple -- you can integrate it using direct formulas of integration.

Answer to 4.2.14 (f) :

Substitute lnx = t then dx=xdt.
Substitution this the integral will look similar to this :

Now, integrate term by term to obtain the answer.

Answer to question 4.2.10

For this question, give the substitution as follows :

Now, differentiating, we obtain the value of dx as follows :

Now substituting these values into the integrand, it becomes as simple as t.dt . Now integrate it and replace t by the substitution we gave above and this yields the answer.