# Integral (Antiderivative) Calculator with Steps

This online calculator will find the indefinite integral (antiderivative) of the given function, with steps shown (if possible).

Enter a function:

Integrate with respect to:

Please write without any differentials such as dx, dy etc.

For definite integral, see definite integral calculator.

Some integrals may take much time. Be patient!

If the integral hasn't been calculated or it took too much time, please write it in comments. The algorithm will be improved.

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## Solution

Your input: find $$\int{x \cos{\left(x^{2} \right)} d x}$$$Let $$u=x^{2}$$$.

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

The integral becomes

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

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

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