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.

Answer to question 4.2.8

Substitute cos x = t. Then, dx = -dt/sinx.
Now the question becomes very simple, try to do the rest by yourself and comment if you need any further help.

Answer to question 4.2.3

For this question, substitute (2x-5) = t and then dx = dt/2
Now the integrand becomes :

After simplification, you obtain :

Now integrating term by term, this yields :

Now all you have to do is replace t with (2x-5) and this will give you the answer.

Answers to question 4.1.22 all subparts

Answer to 4.1.22 (a) :

In this question, we have to transform the denominator and then Integrate term by term as follows :

Now, you can integrate using the standard integration formulas given below :

For the first part use this formula :

and for the second part, use this formula :

 I assume that you now know what  to use as u and a since we've already discussed enough problems like this.

Answer to 4.1.22 (b) :

For this question, I'll give you two hints and that's enough for you to solve it. Use the two hints :
Hint 1, use the below identity :

and then use this :

And now, with the simplified integrand, you can integrate directly. If you have any doubts, leave it in the comments,

Answer to 4.1.21 (c) :

I'll provide full solution to this question. Anyway, here's the hint 10^x = 2^x * 5^x. The transformation are as follows :

We then split the integrand and integrate both parts separately. See the steps below :

Answer to 4.1.22 (d) :

This is a very simple question and can be done directly. Just remember the fact that sin5a is a constant and it's anti-derivative or integral is xsin5a and NOT -(cos5a)/5 !

Answer to question 4.1.21 all subparts

Answer to 4.1.21 (a) :

In this question, we need to transform the denominator in the form (x+m)^2 +n as already discussed while answering the question 4.1.15 .
After transformation, we obtain :

Now the Integrand can be integrated using the standard formula

We have u=(x-3) and a = squareroot(4) = 2. Putting these values in the standard formula, we obtain the answer as :

I use the notation inverse tan instead of arc tan.

Answer to 4.1.21 (b) :

 This is just a normal elementary question, we proceed as follows :

Answer to 4.1.21 (c) : 

Like the previous question, we proceed as follows :

Answer to 4.1.21 (d) :

Split the numerator into 2+2x^2 + x^2 and proceed like the previous questions. Some steps are given below :

Now, all you have to do is expand the integrand into two parts, on the first part (1+x^2) will be canceled and on the second part, x^2 will be canceled. Then, integrate the simplified integrand to obtain the answer :

Answer to question 4.1.20

This is a simple question. We must first make the coefficient of x^2 to 1 and then integrate using the standard formula :

So, Let us first take 10 common from the denominator to make the coefficient of x as 1 :

Now, let's use the standard formula to integrate. We have u=x and a=squareroot(7/10). Putting these values in the standard formula given above will yield this :

After further simplification by dividing root(10) by 10 and taking common denominator inside log, we obtain the answer as :

Please note that I use log instead of ln so in this blog, log means natural log.