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# Solve the trigonometric integral $\int\sec\left(x\right)^4dx$

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##  Final answer to the problem

$\frac{\tan\left(x\right)\sec\left(x\right)^{2}}{3}+\frac{2}{3}\tan\left(x\right)+C_0$
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##  Step-by-step Solution 

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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

Simplify the integral $\int\sec\left(x\right)^4dx$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$\frac{\sin\left(x\right)\sec\left(x\right)^{3}}{3}+\frac{2}{3}\int\sec\left(x\right)^{2}dx$

Learn how to solve trigonometric integrals problems step by step online.

$\frac{\sin\left(x\right)\sec\left(x\right)^{3}}{3}+\frac{2}{3}\int\sec\left(x\right)^{2}dx$

Learn how to solve trigonometric integrals problems step by step online. Solve the trigonometric integral int(sec(x)^4)dx. Simplify the integral \int\sec\left(x\right)^4dx applying the reduction formula, \displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx. The integral \frac{2}{3}\int\sec\left(x\right)^{2}dx results in: \frac{2}{3}\tan\left(x\right). Gather the results of all integrals. Apply the trigonometric identity: \sin\left(\theta \right)\sec\left(\theta \right)^n=\tan\left(\theta \right)\sec\left(\theta \right)^{\left(n-1\right)}, where n=3.

##  Final answer to the problem

$\frac{\tan\left(x\right)\sec\left(x\right)^{2}}{3}+\frac{2}{3}\tan\left(x\right)+C_0$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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9
0
a
b
c
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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

###  Main Topic: Trigonometric Integrals

Integrals that contain trigonometric functions and their powers.