Kalkulator całkujący

Znajdź $\int x \cos{\left(x^{2} \right)}\, dx$.

Let $u=x^{2}$.

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

So,

$${\color{red}{\int{x \cos{\left(x^{2} \right)} d x}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

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 $f{\left(u \right)} = \cos{\left(u \right)}$:

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

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

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{2} = \frac{{\color{red}{\sin{\left(u \right)}}}}{2}$$

Recall that $u=x^{2}$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{2} = \frac{\sin{\left({\color{red}{x^{2}}} \right)}}{2}$$

Therefore,

$$\int{x \cos{\left(x^{2} \right)} d x} = \frac{\sin{\left(x^{2} \right)}}{2}$$

Add the constant of integration:

$$\int{x \cos{\left(x^{2} \right)} d x} = \frac{\sin{\left(x^{2} \right)}}{2}+C$$

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