## Integral Calculus of Substitution Method

Class | Bsc |
---|---|

Topic | Integral Calculus |

Hello friends, welcome to a new post, in today's post, we are going to share some **important question** which is the question of **integral calculus**. All of you will look at this question carefully. If you are facing any problem in understanding this question, then you can take the help of its **video solution**. You will get video solution on YouTube. You will find the **direct link** of the video below the solution. Along with this, you will get the complete solution of the question in the form of an image. If you have any doubt in any question, then you can send your doubt to us. To share doubt, join **WhatsApp** or **Telegram**.

### Question No 1

**1. ****Evaluate integrate (2x ^ 3)/(4 + x ^ 8) dx**

Solution : Evaluate integrate (2x ^ 3)/(4 + x ^ 8) dx To solve this question, first you have to take 2 out of Integrate and now we will write x^8 as (x^4)^2 And now we will put t in place of x^4 as well as differentiate x^4 and t then we will get dx = dt /4x^3. Now let us go back to the question now we will do some changes in this question like 2 integrate (x ^ 3)/(4 + t^2) * dt/ 4x^3 now cut x^3 from x^3 Then we get something like this 2/4 Integrate 1 /(4 + t^2) dt Now we'll write it like this. 2/4 Integrate 1 /(2^2 + t^2) dt Now we will add a formula in this Integrate 1 / x^2 + a^2 = 1 / a tan^-1 x/a then we will get | 1 / 2 tan^-1 t /2 Now we will put x^4 in place of t again, then we will get the answer which will be something like this. 1 / 2 tan^-1 x^4 /2 + c.

Aman Kumar helped us in solving this question. That's why I express my gratitude to you very much. And if you also have any doubt, then you can share that doubt with us. We will try to reach its solution to you as soon as possible.

Solution Video On Youtube

### Question No 2

**2. ****Evaluate integrate (2x + 3)/(x ^ 2 + 3x + 7) dx**

Solution : Evaluate integrate (2x + 3)/(x ^ 2 + 3x + 7) dx To solve this question, first we will put t in place of x ^ 2 + 3x + 7 as well as differentiate x ^ 2 + 3x + 7 and t then we will get dx = dt /2x + 3. Now let us go back to the question now we will do some changes in this question like integrate (2x + 3)/t * dt/ 2x + 3 now cut 2x + 3 from 2x + 3. Then we get something like this Integrate 1/t dt Now we will add a formula in this Integrate 1 / x dx = log(x) + c then we will get | log(t) + c Now we will put x^2 + 3x + 7 in place of t again, then we will get the answer which will be something like this. log(x^2 + 3x + 7) + c.

And if you also have any doubt, then you can share that doubt with us. We will try to reach its solution to you as soon as possible.

Solution Video On Youtube

### Question No 3

**3. ****Evaluate integrate (1 - sin x)/(x + cos x) dx**

Solution : Evaluate integrate (1 - sin x)/(x + cos x) dx To solve this question, first we will put t in place of x + cos x as well as differentiate x + cos x and t then we will get dx = dt /1 - sin x. Now let us go back to the question now we will do some changes in this question like integrate (1 - sin x)/t * dt/ (1 - sin x) now cut (1 - sin x) from (1 - sin x). Then we get something like this Integrate 1/t dt Now we will add a formula in this Integrate 1 / x dx = log(x) + c then we will get | log(t) + c Now we will put x^2 + 3x + 7 in place of t again, then we will get the answer which will be something like this. log(x + cos x) + c.

And if you also have any doubt, then you can share that doubt with us. We will try to reach its solution to you as soon as possible.

Solution Video On Youtube

### Question No 4

**4. ****Evaluate integrate (x ^ (n - 1))/(1 + x ^ n) dx**

Solution : Evaluate integrate (x ^ (n - 1))/(1 + x^n) dx To solve this question, first we will put t in place of 1 + x^n as well as differentiate 1 + x^n and t then we will get dx = dt /nx^n - 1. Now let us go back to the question now we will do some changes in this question like integrate (x ^ (n - 1))/t * dt/ (x ^ (n - 1)) now cut(x ^ (n - 1)) from (x ^ (n - 1)). Then we get something like this 1/n Integrate 1/t dt Now we will add a formula in this Integrate 1 / x dx = log(x) + c then we will get | 1/n * log(t) + c Now we will put 1 + x^n in place of t again, then we will get the answer which will be something like this. log(1 + x^n ) + c.

And if you also have any doubt, then you can share that doubt with us. We will try to reach its solution to you as soon as possible.

Solution Video On Youtube

*Aman Kumar helped us in solving this question. That's why I express my gratitude to you very much.*

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