Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- 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
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Applying the trigonometric identity: $\sin\left(\theta \right)^2 = 1-\cos\left(\theta \right)^2$
Multiplying polynomials $\cos\left(x\right)^2$ and $1-\cos\left(x\right)^2$
Any expression multiplied by $1$ is equal to itself
When multiplying two powers that have the same base ($\cos\left(x\right)^2$), you can add the exponents
Simplify $\left(\cos\left(x\right)^2\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $2$
Multiply $2$ times $2$
Multiply $2$ times $2$
Rewrite the trigonometric expression $\cos\left(x\right)^2\sin\left(x\right)^2$ inside the integral
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
Apply the formula: $\int\cos\left(\theta \right)^2dx$$=\frac{1}{2}\theta +\frac{1}{4}\sin\left(2\theta \right)+C$
The integral $\int\cos\left(x\right)^2dx$ results in: $\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)$
Gather the results of all integrals
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Apply the formula: $\int\cos\left(\theta \right)^ndx$$=\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 $n=4$
Solve the product
Apply the formula: $\int\cos\left(\theta \right)^2dx$$=\frac{1}{2}\theta +\frac{1}{4}\sin\left(2\theta \right)+C$
The integral $\int-\cos\left(x\right)^{4}dx$ results in: $\frac{-\cos\left(x\right)^{3}\sin\left(x\right)}{4}-\frac{3}{4}\left(\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)\right)$
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 $C$
Solve the product $-\frac{3}{4}\left(\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)\right)$
Combining like terms $\frac{1}{4}\sin\left(2x\right)$ and $-\frac{3}{4}\cdot \frac{1}{4}\sin\left(2x\right)$
Divide $-2$ by $4$
Multiplying fractions $-\frac{3}{4} \times \frac{1}{2}$
Expand and simplify
