Test: Integrals Of Trigonometric Identities - JEE MCQ

# Test: Integrals Of Trigonometric Identities - JEE MCQ

Test Description

## 5 Questions MCQ Test Mathematics (Maths) Class 12 - Test: Integrals Of Trigonometric Identities

Test: Integrals Of Trigonometric Identities for JEE 2024 is part of Mathematics (Maths) Class 12 preparation. The Test: Integrals Of Trigonometric Identities questions and answers have been prepared according to the JEE exam syllabus.The Test: Integrals Of Trigonometric Identities MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Integrals Of Trigonometric Identities below.
Solutions of Test: Integrals Of Trigonometric Identities questions in English are available as part of our Mathematics (Maths) Class 12 for JEE & Test: Integrals Of Trigonometric Identities solutions in Hindi for Mathematics (Maths) Class 12 course. Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free. Attempt Test: Integrals Of Trigonometric Identities | 5 questions in 10 minutes | Mock test for JEE preparation | Free important questions MCQ to study Mathematics (Maths) Class 12 for JEE Exam | Download free PDF with solutions
Test: Integrals Of Trigonometric Identities - Question 1

### The integral of tan4x is:

Detailed Solution for Test: Integrals Of Trigonometric Identities - Question 1

Begin by rewriting  ∫tan4xdx as ∫tan2xtan2xdx.

Now we can apply the Pythagorean Identity,  tan2x+1=sec2x,                                         or tan2x=sec2x−1

∫tan2x tan2x dx = ∫(sec2x−1)tan2xdx

Distributing the tan2x:
= ∫sec2xtan2x − tan2xdx
Applying the sum rule:
= ∫sec2xtan2xdx − ∫tan2xdx

We'll evaluate these integrals one by one.

First Integral
This one is solved using a
Let u = tanx

Applying the substitution,
Because u = tanx,

Second Integral
Since we don't really know what  ∫tan2xdx is by just looking at it, try applying the tan2x = sec2x−1

identity again:

∫tan2xdx = ∫(sec2x−1)dx

Using the sum rule, the integral boils down to:
∫sec2xdx − ∫1dx

The first of these,  ∫sec2xdx, is just tanx + C.

The second one, the so-called "perfect integral", is simply x+C.

Putting it all together, we can say:
∫tan2xdx = tanx + C − x + C

And because C+C is just another arbitrary constant, we can combine it into a general constant C:
∫tan2xdx = tanx − x + C

Combining the two results, we have:
∫tan4xdx=∫sec2xtan2xdx−∫tan2xdx

=(tan3x/3 + C) − (tanx − x + C)

=tan3x/3 − tanx + x + C

Again, because C+C is a constant, we can join them into one C.

Test: Integrals Of Trigonometric Identities - Question 2

### Integrate

Detailed Solution for Test: Integrals Of Trigonometric Identities - Question 2

∫(2+tan x)2dx
= ∫(4 + tan2 x + 4tan x)dx
= ∫4 dx + ∫tan2 x dx + 4∫tan x dx
= 4x + ∫(sec2 x - 1)dx + 4(log|sec x|)
= 4x + tanx - x + 4(log|sec x|)
3x + tanx + 4(log|sec x|) + c

 1 Crore+ students have signed up on EduRev. Have you?
Test: Integrals Of Trigonometric Identities - Question 3

### Evaluate:

Detailed Solution for Test: Integrals Of Trigonometric Identities - Question 3

∫sin2(2x+1) dx
Put t = 2x+1
dt = 2dx
dx= dt/2
= 1/2∫sin2 t dt
=1/2∫(1-cos2t)/2 dt
= 1/4∫(1-cos2t) dt
= ¼[(t - (sin2t)/2]dt
= t/4 - sin2t/8 + c
= (2x+1)/4 - ⅛(sin(4x+2)) + c
= x/2 - 1/8sin(4x+2) + ¼ + c
As ¼ is also a constant, so eq is = x/2 - 1/8sin(4x+2) + c

Test: Integrals Of Trigonometric Identities - Question 4

The value of  bb

Detailed Solution for Test: Integrals Of Trigonometric Identities - Question 4

Let cos−1(sinx)=θ
⇒ sinx=cosθ
⇒ sinx=sin(π/2−θ)
⇒ x = π/2−θ
⇒ θ = π/2−x
∴ ∫cos−1(sinx)dx=∫(π/2−x)dx
= ∫π/2dx−∫xdx
= πx/2 - x2/2 + c, where b is a constant of integration.

Test: Integrals Of Trigonometric Identities - Question 5

Detailed Solution for Test: Integrals Of Trigonometric Identities - Question 5

I = ∫cos2x/(sinx+cosx)2dx
⇒I = ∫cos2x−sin2x(sinx+cosx)2dx
⇒I = ∫[(cosx+sinx)(cosx−sinx)]/(sinx+cosx)2dx
⇒I = ∫(cosx−sinx)/(sinx+cosx)dx
Let sinx+cosx = t
(cosx−sinx)dx = dt
Then, I = ∫dt/t
I = log|t|+c
I = log|sinx + cosx| + c

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests
Information about Test: Integrals Of Trigonometric Identities Page
In this test you can find the Exam questions for Test: Integrals Of Trigonometric Identities solved & explained in the simplest way possible. Besides giving Questions and answers for Test: Integrals Of Trigonometric Identities, EduRev gives you an ample number of Online tests for practice

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests