Integral Calculus of Substitution Method Question With Solution Download Pdf

Integral Calculus of Substitution Method 

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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. 

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Solution Shared By :- Aman Kumar

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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 Shared By :- Aman Kumar

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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 Shared By :- Aman Kumar

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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 Shared By :- Aman Kumar

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