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Solve the trigonometric integral $\int\left(\sec\left(x\right)^6-\sec\left(x\right)^4\right)dx$

Step-by-step Solution

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Final Answer

$-\frac{2}{15}\tan\left(x\right)+\frac{4\tan\left(x\right)\sec\left(x\right)^{2}}{15}+\frac{\tan\left(x\right)\sec\left(x\right)^{4}}{5}+\frac{-\sin\left(x\right)\sec\left(x\right)^{3}}{3}+C_0$
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Step-by-step Solution

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Expand the integral $\int\left(\sec\left(x\right)^6-\sec\left(x\right)^4\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int\sec\left(x\right)^6dx+\int-\sec\left(x\right)^4dx$

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

$\int\sec\left(x\right)^6dx+\int-\sec\left(x\right)^4dx$

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Learn how to solve trigonometric integrals problems step by step online. Solve the trigonometric integral int(sec(x)^6-sec(x)^4)dx. Expand the integral \int\left(\sec\left(x\right)^6-\sec\left(x\right)^4\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\sec\left(x\right)^6dx results in: \frac{\tan\left(x\right)\sec\left(x\right)^{4}}{5}+\frac{4\tan\left(x\right)\sec\left(x\right)^{2}}{15}+\frac{8}{15}\tan\left(x\right). Gather the results of all integrals. The integral \int-\sec\left(x\right)^4dx results in: \frac{-\sin\left(x\right)\sec\left(x\right)^{3}}{3}-\frac{2}{3}\tan\left(x\right).

Final Answer

$-\frac{2}{15}\tan\left(x\right)+\frac{4\tan\left(x\right)\sec\left(x\right)^{2}}{15}+\frac{\tan\left(x\right)\sec\left(x\right)^{4}}{5}+\frac{-\sin\left(x\right)\sec\left(x\right)^{3}}{3}+C_0$

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

Solve integral of (secx^6+-1secx^4)dx using basic integralsSolve integral of (secx^6+-1secx^4)dx using u-substitutionSolve integral of (secx^6+-1secx^4)dx using integration by parts

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Function Plot

Plotting: $-\frac{2}{15}\tan\left(x\right)+\frac{4\tan\left(x\right)\sec\left(x\right)^{2}}{15}+\frac{\tan\left(x\right)\sec\left(x\right)^{4}}{5}+\frac{-\sin\left(x\right)\sec\left(x\right)^{3}}{3}+C_0$

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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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Main Topic: Trigonometric Integrals

Integrals that contain trigonometric functions and their powers.

Used Formulas

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