Integral (Antiderivative) Calculator with Steps

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

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For definite integral, see definite integral calculator.

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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 $$$x dx = \frac{du}{2}$$$.

Thus,

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

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