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

The value of etanθ(secθsinθ) dθ is equal to
(a) etanθsinθ+C
(b) etanθsinθ+C
(c) etanθsecθ+C
(d) etanθcosθ+C

Answer
VerifiedVerified
431.4k+ views
like imagedislike image
Hint: In this type of question we have to use the concept of integration by parts. We know that when we have to integrate a product of two functions at that time we use integration by parts. We can write the formula of integration of by parts as, uvdx=uvdx[dudxvdx]dx. In this case we have to select the first function that is u in such a way that the derivative of the function could be easily integrated. In the given example we first separate out the integral over subtraction and then we will use the formula of integration by parts.

Complete step by step answer:
Now we have to find the value of the integral etanθ(secθsinθ) dθ.
Let us first separate out the integral over subtraction
etanθ(secθsinθ) dθetanθsecθdθetanθsinθdθ
Now, here we get two integrals to simplify it further. We kept the first integral as it is and we will evaluate the second integral by using the integration by parts.
We know that the formula of integration by parts is given by,
uvdx=uvdx[dudxvdx]dx
In the second integral let us consider u=etanθ,v=sinθ. Thus by using above formula we can write,
etanθsecθdθ[etanθsinθdθ(ddθetanθsinθdθ) dθ]
As we know that, sinθdθ=cosθ and ddθ(etanθ)=etanθsec2θ
etanθsecθdθ[etanθ(cosθ)etanθsec2θ(cosθ)dθ]
Also we know that, secθ=1cosθ. Hence, by simplifying the expression further we can write,
etanθsecθdθ[etanθcosθ+etanθsecθdθ]etanθsecθdθ+etanθcosθetanθsecθdθ
Here, we can observe that first and third terms are equal having opposite signs so they get cancel with each other
etanθcosθ+C
Hence, the value of etanθ(secθsinθ) dθ is etanθcosθ+C

So, the correct answer is “Option d”.

Note: In this type of question students have to remember to first separate the integral over subtraction and then use the formula of integration by parts. Also students have to note that if trigonometric functions are present and integration by parts is applicable then that trigonometric function always appears as a second function. Students have to take care in calculation of derivatives of (etanθ).