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$\int\frac{1}{1+\cos\left(x\right)^2}dx$

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Solution

Final answer to the problem

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

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We can solve the integral $\int\frac{1}{1+\cos\left(x\right)^2}dx$ 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

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

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

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Learn how to solve problems step by step online. int(1/(1+cos(x)^2))dx. We can solve the integral \int\frac{1}{1+\cos\left(x\right)^2}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. Hence. Substituting in the original integral we get. Simplifying.

Final answer to the problem

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

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Function Plot

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

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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
D□x
∫
∫◻
|◻|
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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