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.

Answer to Question 4.1.18

The denominator of this question needs to be transformed in the form (x+m)^2 + n (this form will be inside the square root) (please see question 4.1.15 here to know how you can find the values of m and n) and then we can use the standard integration formula :

After choosing values of m and n, the transformed equation looks like this :

Now we can use the standard integration formula. We have u=(x+3) and a = squareroot(8). This yields the answer :

NOTE : I use log instead of ln most of the time. So when I use log, consider it as ln(i.e natural log of base e).

Answer to question 4.1.15

There is a trick for doing this type of question having denominator of integrand in the form ax^2 + bx + c :

We transform the  denominator in the form (x + m)^2 + n and the first thing you have to do is decide m and this is how you do it : select m in such a way that when you expand (x+m)^2, the coefficient of x will be equal to the coefficient of x in the equation ax^2 + bx + c and then select n so that the constant term while expanding (x+m)^2 is equal to c  in the equation.

Here's how we transform the integrand using the above method :

On the above step, notice how we chose 1/2 and 3/4 as m and n. Now, the integrand can be transformed to use the standard integration formula

This is how we transform our integrand :

Now, let's use the standard formula. We have our u=(x+1/2) and a =square_root(3)/2 . This gives us :

 

Now, after simplification, we obtain the answer :