Integral Calculator

Find indefinite integrals (antiderivatives) step by step

This online calculator will try to find the indefinite integral (antiderivative) of the given function, with steps shown. Different techniques are used: integration by substitution, integration by parts, integration by partial fractions, trigonometric substitutions, etc.

Related calculator: Definite and Improper Integral Calculator

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Your Input

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

Solution

Let $$$u=x^{2}$$$.

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

Therefore,

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

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

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