Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

What is 01tan1x1+x2dx equal to?
A.π28
B.π232
C.π4
D.π2

Answer
VerifiedVerified
425.1k+ views
like imagedislike image
Hint: Here in this question, we have to find the integrated value of a given trigonometric function. This can be solved by using a substitution method and later integrated by using the standard trigonometric formula of integration. Since they have mentioned the limit points. It is a definite integral on applying the limits we get the required solution.

Complete step by step solution:
In integration we have two different kinds. One is definite integral and another one is indefinite integral. In definite integral the limits points are mentioned. In indefinite integral the limit points are not mentioned.
Here this question belongs to the definite integral where the limits points are mentioned.
Now consider the given function
01tan1x1+x2dx--------(1)
Here the lower limit is 0 and the upper limit is 1
We integrate the above function by using a substitution method
Let take, t=tan1x
Differentiate with respect to x, then
dtdx=11+x2
Or
dt=11+x2dx
Substitute t and dt in equation (1), we have
 01tdt
Now integrate using a formula abxn=xn+1n+1|ab, then we gave
t22|01
Put, t=tan1x, then
 12(tan1x)2|01
On applying a limit, we have
12[(tan11)2(tan10)2]
As we know the value of tan11=π4 and tan10=0, then on substituting the values we get
12[(π4)2(0)2]
On simplification, we get
12[(π4)2]
12[π216]
π232
Hence, we have integrated the given function and applied the limit points and obtained an answer.
The value of 01tan1x1+x2dx=π232. So, option (B) is correct.

Note: By simplifying the question using the substitution we can integrate the given function easily. If we apply integration directly it may be complicated to solve further. So, simplification is needed. We must know the differentiation and integration formulas. The standard integration formulas for the trigonometric ratios must know.