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Solve the trigonometric integral $\int\frac{1}{\cos\left(x\right)-1}dx$

Step-by-step Solution

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Final answer to the problem

$\tan\left(\frac{x}{2}\right)^{-1}+C_0$
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Step-by-step Solution

How should I solve this problem?

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  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
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1

We can solve the integral $\int\frac{1}{\cos\left(x\right)-1}dx$ by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of $t$ by setting the substitution

$t=\tan\left(\frac{x}{2}\right)$
2

Hence

$\sin x=\frac{2t}{1+t^{2}},\:\cos x=\frac{1-t^{2}}{1+t^{2}},\:\mathrm{and}\:\:dx=\frac{2}{1+t^{2}}dt$
3

Substituting in the original integral we get

$\int\frac{1}{\frac{1-t^{2}}{1+t^{2}}-1}\frac{2}{1+t^{2}}dt$
4

Simplifying

$\int\frac{1}{-t^{2}}dt$
5

Take the constant $\frac{1}{-1}$ out of the integral

$-\int\frac{1}{t^{2}}dt$
6

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$-\int t^{-2}dt$
7

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$

$\frac{-t^{-1}}{-1}$
8

Simplify the fraction $\frac{-t^{-1}}{-1}$ by $-1$

$t^{-1}$
9

Replace $t$ with the value that we assigned to it in the beginning: $\tan\left(\frac{x}{2}\right)$

$\tan\left(\frac{x}{2}\right)^{-1}$
10

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\tan\left(\frac{x}{2}\right)^{-1}+C_0$

Final answer to the problem

$\tan\left(\frac{x}{2}\right)^{-1}+C_0$

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Function Plot

Plotting: $\tan\left(\frac{x}{2}\right)^{-1}+C_0$

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1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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