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Exercise

$\int\left(\cos\left(x\right)^2\cdot\sin\left(x\right)^2\right)dx$

Step-by-step Solution

1

Rewrite the trigonometric expression $\cos\left(x\right)^2\sin\left(x\right)^2$ inside the integral

$\int\left(\cos\left(x\right)^2-\cos\left(x\right)^{4}\right)dx$
2

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

$\int\cos\left(x\right)^2dx+\int-\cos\left(x\right)^{4}dx$
3

The integral $\int\cos\left(x\right)^2dx$ results in: $\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)$

$\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)$
4

Gather the results of all integrals

$\frac{1}{4}\sin\left(2x\right)+\frac{1}{2}x+\int-\cos\left(x\right)^{4}dx$
5

The integral $\int-\cos\left(x\right)^{4}dx$ results in: $\frac{-\cos\left(x\right)^{3}\sin\left(x\right)}{4}-\frac{3}{4}\left(\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)\right)$

$\frac{-\cos\left(x\right)^{3}\sin\left(x\right)}{4}-\frac{3}{4}\left(\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)\right)$
6

Gather the results of all integrals

$\frac{1}{4}\sin\left(2x\right)+\frac{1}{2}x-\frac{3}{4}\left(\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)\right)+\frac{-\cos\left(x\right)^{3}\sin\left(x\right)}{4}$
7

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{4}\sin\left(2x\right)+\frac{1}{2}x-\frac{3}{4}\left(\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)\right)+\frac{-\cos\left(x\right)^{3}\sin\left(x\right)}{4}+C_0$
8

Expand and simplify

$-\frac{1}{2}\sin\left(2x\right)+\frac{1}{2}x-\frac{3}{8}x+\frac{-\cos\left(x\right)^{3}\sin\left(x\right)}{4}+C_0$

Final answer to the exercise

$-\frac{1}{2}\sin\left(2x\right)+\frac{1}{2}x-\frac{3}{8}x+\frac{-\cos\left(x\right)^{3}\sin\left(x\right)}{4}+C_0$

Try other ways to solve this exercise

  • Integrate using basic integrals
  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Product of Binomials with Common Term
  • FOIL Method
  • Load more...
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Similar Questions Solved

$\frac{dy}{dx}=\tan\left(x+y\right)$

Main Topic: Integrals with Radicals

Integrals with radicals are those integrals that contain a radical (square root, cubic, etc.) in the numerator or denominator of the integral.

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