# Integral de $\cos^{4}{\left(x \right)}$

La calculadora encontrará la integral/antiderivada de $\cos^{4}{\left(x \right)}$, con los pasos que se muestran.

Escriba sin diferenciales como $dx$, $dy$ etc.
Deje vacío para la detección automática.

Si la calculadora no calculó algo o ha identificado un error, o tiene una sugerencia/comentario, escríbalo en los comentarios a continuación.

### Tu aportación

Encuentra $\int \cos^{4}{\left(x \right)}\, dx$.

### Solución

Rewrite the cosine using the power reducing formula $\cos^{4}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{\cos{\left(4 \alpha \right)}}{8} + \frac{3}{8}$ with $\alpha=x$:

$$\color{red}{\int{\cos^{4}{\left(x \right)} d x}} = \color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{8} + \frac{3}{8}\right)d x}}$$

Apply the constant multiple rule $\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$ with $c=\frac{1}{8}$ and $f{\left(x \right)} = 4 \cos{\left(2 x \right)} + \cos{\left(4 x \right)} + 3$:

$$\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{\cos{\left(4 x \right)}}{8} + \frac{3}{8}\right)d x}} = \color{red}{\left(\frac{\int{\left(4 \cos{\left(2 x \right)} + \cos{\left(4 x \right)} + 3\right)d x}}{8}\right)}$$

Integrate term by term:

$$\frac{\color{red}{\int{\left(4 \cos{\left(2 x \right)} + \cos{\left(4 x \right)} + 3\right)d x}}}{8} = \frac{\color{red}{\left(\int{3 d x} + \int{4 \cos{\left(2 x \right)} d x} + \int{\cos{\left(4 x \right)} d x}\right)}}{8}$$

Apply the constant rule $\int c\, dx = c x$ with $c=3$:

$$\frac{\int{4 \cos{\left(2 x \right)} d x}}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\int{3 d x}}}{8} = \frac{\int{4 \cos{\left(2 x \right)} d x}}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\left(3 x\right)}}{8}$$

Apply the constant multiple rule $\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$ with $c=4$ and $f{\left(x \right)} = \cos{\left(2 x \right)}$:

$$\frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\int{4 \cos{\left(2 x \right)} d x}}}{8} = \frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\left(4 \int{\cos{\left(2 x \right)} d x}\right)}}{8}$$

Let $u=2 x$.

Then $du=\left(2 x\right)^{\prime }dx = 2 dx$ (steps can be seen here), and we have that $dx = \frac{du}{2}$.

The integral becomes

$$\frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\int{\cos{\left(2 x \right)} d x}}}{2} = \frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}{2}$$

Apply the constant multiple rule $\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$ with $c=\frac{1}{2}$ and $f{\left(u \right)} = \cos{\left(u \right)}$:

$$\frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}{2} = \frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}{2}$$

The integral of the cosine is $\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$:

$$\frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\int{\cos{\left(u \right)} d u}}}{4} = \frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\color{red}{\sin{\left(u \right)}}}{4}$$

Recall that $u=2 x$:

$$\frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\sin{\left(\color{red}{u} \right)}}{4} = \frac{3 x}{8} + \frac{\int{\cos{\left(4 x \right)} d x}}{8} + \frac{\sin{\left(\color{red}{\left(2 x\right)} \right)}}{4}$$

Let $u=4 x$.

Then $du=\left(4 x\right)^{\prime }dx = 4 dx$ (steps can be seen here), and we have that $dx = \frac{du}{4}$.

Therefore,

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\color{red}{\int{\cos{\left(4 x \right)} d x}}}{8} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}{8}$$

Apply the constant multiple rule $\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$ with $c=\frac{1}{4}$ and $f{\left(u \right)} = \cos{\left(u \right)}$:

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}{8} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}{8}$$

The integral of the cosine is $\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$:

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\color{red}{\int{\cos{\left(u \right)} d u}}}{32} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\color{red}{\sin{\left(u \right)}}}{32}$$

Recall that $u=4 x$:

$$\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(\color{red}{u} \right)}}{32} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(\color{red}{\left(4 x\right)} \right)}}{32}$$

Therefore,

$$\int{\cos^{4}{\left(x \right)} d x} = \frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32}$$

Simplify:

$$\int{\cos^{4}{\left(x \right)} d x} = \frac{12 x + 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32}$$

$$\int{\cos^{4}{\left(x \right)} d x} = \frac{12 x + 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32}+C$$
Answer: $\int{\cos^{4}{\left(x \right)} d x}=\frac{12 x + 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32}+C$