
-What is the integral of ?
Answer
416.7k+ views
Hint: To solve this type of question we use concepts of integration without limits .Let us substitute in the given integral to find the value of the given integral easily. So, let us assume the denominator as t and then substitute the numerator in terms of dt.
Complete step-by-step answer:
Now, we will use the substitution technique. We will let the denominator term as t and then differentiate the denominator term because the given integral is in proper form i.e. the degree of numerator is less than degree of denominator. So,
Let
Differentiating both sides with respect to x, we get
(As we know for and for other term on L.H.S )
Substituting the value of t and dx in the given integral and we get,
Now splitting this we can write as follows
Now as we use the formula
where is the integration constant
Let us take
We get
Now putting the value of t in the above equation, we get
So, the correct answer is “ ”.
Note: While solving questions which include integration of given terms, we have to check whether the given integral is proper or improper. In a proper integral, the degree of the numerator is less than that of the denominator and vice – versa in the improper integral. Also, we have to write the integration constant c when we are dealing with indefinite integrals i.e. integrals with no limit.
Complete step-by-step answer:
Now, we will use the substitution technique. We will let the denominator term as t and then differentiate the denominator term because the given integral is in proper form i.e. the degree of numerator is less than degree of denominator. So,
Let
Differentiating both sides with respect to x, we get
(As we know
Substituting the value of t and dx in the given integral and
Now splitting this we can write as follows
Now as we use the formula
Let us take
We get
Now putting the value of t in the above equation, we get
So, the correct answer is “
Note: While solving questions which include integration of given terms, we have to check whether the given integral is proper or improper. In a proper integral, the degree of the numerator is less than that of the denominator and vice – versa in the improper integral. Also, we have to write the integration constant c when we are dealing with indefinite integrals i.e. integrals with no limit.
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