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Solve the trigonometric integral $\int\left(\sin\left(x\right)\cos\left(x\right)\right)^4dx$

Step-by-step Solution

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Final Answer

$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)^{5}}{8}+\frac{3}{128}x+\frac{3}{256}\sin\left(2x\right)-\frac{1}{16}\cos\left(x\right)^{5}\sin\left(x\right)+\frac{1}{64}\cos\left(x\right)^{3}\sin\left(x\right)+C_0$
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Step-by-step Solution

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The power of a product is equal to the product of it's factors raised to the same power

$\int\sin\left(x\right)^4\cos\left(x\right)^4dx$

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

$\int\sin\left(x\right)^4\cos\left(x\right)^4dx$

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Learn how to solve trigonometric integrals problems step by step online. Solve the trigonometric integral int((sin(x)cos(x))^4)dx. The power of a product is equal to the product of it's factors raised to the same power. Apply the formula: \int\sin\left(\theta \right)^n\cos\left(\theta \right)^mdx=\frac{-\sin\left(\theta \right)^{\left(n-1\right)}\cos\left(\theta \right)^{\left(m+1\right)}}{n+m}+\frac{n-1}{n+m}\int\sin\left(\theta \right)^{\left(n-2\right)}\cos\left(\theta \right)^mdx, where m=4 and n=4. Simplify the expression inside the integral. The integral \frac{3}{8}\int\sin\left(x\right)^{2}\cos\left(x\right)^4dx results in: \frac{1}{64}\cos\left(x\right)^{3}\sin\left(x\right)+\frac{9}{64}x+\frac{9}{128}\sin\left(2x\right)-\frac{1}{16}\cos\left(x\right)^{5}\sin\left(x\right)-\frac{15}{64}\left(\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)\right).

Final Answer

$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)^{5}}{8}+\frac{3}{128}x+\frac{3}{256}\sin\left(2x\right)-\frac{1}{16}\cos\left(x\right)^{5}\sin\left(x\right)+\frac{1}{64}\cos\left(x\right)^{3}\sin\left(x\right)+C_0$

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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 (sinxcosx^4)dx using basic integralsSolve integral of (sinxcosx^4)dx using u-substitutionSolve integral of (sinxcosx^4)dx using integration by parts

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Function Plot

Plotting: $\frac{-\sin\left(x\right)^{3}\cos\left(x\right)^{5}}{8}+\frac{3}{128}x+\frac{3}{256}\sin\left(2x\right)-\frac{1}{16}\cos\left(x\right)^{5}\sin\left(x\right)+\frac{1}{64}\cos\left(x\right)^{3}\sin\left(x\right)+C_0$

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a
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x
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◻/◻
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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

How to improve your answer:

Main Topic: Trigonometric Integrals

Integrals that contain trigonometric functions and their powers.

Used Formulas

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