Mathematics

Evaluate: $$\displaystyle \int \dfrac{dx}{1-\tan x} $$


SOLUTION
Solution:
$$I=\displaystyle \int \dfrac {dx}{1-\tan x}$$

$$=\displaystyle \int \dfrac {\cos x}{\cos -\sin x}dx$$

$$=\displaystyle \int \dfrac {2\cos x}{2(\cos x-\sin x)}dx$$

$$=\dfrac {1}{2}\displaystyle \int \dfrac {\cos x+\cos x+\sin x-\sin x}{\cos x-\sin x}dx$$

$$=\dfrac {1}{2}\displaystyle \int \left (\dfrac {\cos x-\sin x}{\cos x-\sin x}+\dfrac {\cos x+\sin x}{\cos x-\sin x}\right)dx$$

$$=\dfrac {1}{2} \displaystyle \int 1dx+1 \dfrac {\cos x+\sin x}{\cos x-\sin x}dx$$

Let $$\cos x -\sin x=t$$

$$\Rightarrow \ (-\sin x-\cos x)dx=dt$$

$$\Rightarrow \ -(\sin x+\cos x)dx=dt$$

$$\Rightarrow \ (\sin x+\cos x)dx=dt$$

$$I=\dfrac {1}{2} \left [x+\displaystyle \int {-\dfrac {dt}{t}}\right]$$

$$=\dfrac {1}{2}[x-\log |t|]+C$$

$$=\dfrac {1}{2}[x-\log |\cos x-\sin x|]+C$$
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Evaluate :
$$\displaystyle \int (|x - 1| + |x - 2| + |x - 4|)dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
If $$\int \left( x ^ { 6 } + 7 x ^ { 5 } + 6 x ^ { 4 } + 5 x ^ { 3 } + 4 x ^ { 2 } + 3 x + 1 \right) e ^ { x } d x$$ is equal to $$\sum _ { k = 1 } ^ { \infty } \beta x ^ { k } \cdot e ^ { x } + c$$  (where C is constant of integration) then $$( \alpha + \beta )$$ is-  
  • A. $$8$$
  • B. $$9$$
  • C. $$10$$
  • D. $$7$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
Evaluate: $$\displaystyle \int_{-2}^{2}(x^{11}\cos x+e^{x})dx$$
  • A. $$\sinh 2$$
  • B. $$\displaystyle \frac{3}{2} \sinh 2$$
  • C. $$\displaystyle \frac{\sinh2}{2}$$
  • D. $$ 2\sinh 2$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Hard

$$I:\displaystyle \int\frac{a\cos x+b\sin x}{c\cos x+d\sin x}dx=$$$$\displaystyle \frac{ac+bd}{c^{2}+d^{2}}x+\frac{ad-bc}{c^{2}+d^{2}}\log|c\cos x+d\sin x|+k$$
$$II:\displaystyle \int\frac{\cos x+\sin x}{\sin x-\cos x}dx=\log|\sin x-\cos x|+k$$ Which of the following is true
  • A. Only I
  • B. Only II
  • C. neither I nor II are true
  • D. Both I and II

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Easy
Evaluate:
$$ \int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer