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# Solve the trigonometric integral $\int\cos\left(x\right)^6dx$

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##  Final answer to the problem

$\frac{\cos\left(x\right)^{5}\sin\left(x\right)}{6}+\frac{5}{16}x+\frac{5}{32}\sin\left(2x\right)+\frac{5\cos\left(x\right)^{3}\sin\left(x\right)}{24}+C_0$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• 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

Apply the formula: $\int\cos\left(\theta \right)^ndx$$=\frac{\cos\left(\theta \right)^{\left(n-1\right)}\sin\left(\theta \right)}{n}+\frac{n-1}{n}\int\cos\left(\theta \right)^{\left(n-2\right)}dx$, where $n=6$

$\frac{\cos\left(x\right)^{5}\sin\left(x\right)}{6}+\frac{5}{6}\int\cos\left(x\right)^{4}dx$

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$\frac{\cos\left(x\right)^{5}\sin\left(x\right)}{6}+\frac{5}{6}\int\cos\left(x\right)^{4}dx$

Learn how to solve problems step by step online. Solve the trigonometric integral int(cos(x)^6)dx. Apply the formula: \int\cos\left(\theta \right)^ndx=\frac{\cos\left(\theta \right)^{\left(n-1\right)}\sin\left(\theta \right)}{n}+\frac{n-1}{n}\int\cos\left(\theta \right)^{\left(n-2\right)}dx, where n=6. The integral \frac{5}{6}\int\cos\left(x\right)^{4}dx results in: \frac{5\cos\left(x\right)^{3}\sin\left(x\right)}{24}+\frac{5}{8}\left(\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)\right). Gather the results of all integrals. As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration C.

##  Final answer to the problem

$\frac{\cos\left(x\right)^{5}\sin\left(x\right)}{6}+\frac{5}{16}x+\frac{5}{32}\sin\left(2x\right)+\frac{5\cos\left(x\right)^{3}\sin\left(x\right)}{24}+C_0$

##  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

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0
a
b
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f
g
m
n
u
v
w
x
y
z
.
(◻)
+
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×
◻/◻
/
÷
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