Integral of -sqrt(x)*tan(1)

The calculator will find the integral/antiderivative of $$$- \sqrt{x} \tan{\left(1 \right)}$$$, with steps shown.

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Find $$$\int \left(- \sqrt{x} \tan{\left(1 \right)}\right)\, dx$$$.

The trigonometric functions expect the argument in radians. To enter the argument in degrees, multiply it by pi/180, e.g. write 45° as 45*pi/180, or use the appropriate function adding 'd', e.g. write sin(45°) as sind(45).

Solution

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=- \tan{\left(1 \right)}$$$ and $$$f{\left(x \right)} = \sqrt{x}$$$:

$${\color{red}{\int{\left(- \sqrt{x} \tan{\left(1 \right)}\right)d x}}} = {\color{red}{\left(- \tan{\left(1 \right)} \int{\sqrt{x} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{1}{2}$$$:

$$- \tan{\left(1 \right)} {\color{red}{\int{\sqrt{x} d x}}}=- \tan{\left(1 \right)} {\color{red}{\int{x^{\frac{1}{2}} d x}}}=- \tan{\left(1 \right)} {\color{red}{\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}=- \tan{\left(1 \right)} {\color{red}{\left(\frac{2 x^{\frac{3}{2}}}{3}\right)}}$$

Therefore,

$$\int{\left(- \sqrt{x} \tan{\left(1 \right)}\right)d x} = - \frac{2 x^{\frac{3}{2}} \tan{\left(1 \right)}}{3}$$

Add the constant of integration:

$$\int{\left(- \sqrt{x} \tan{\left(1 \right)}\right)d x} = - \frac{2 x^{\frac{3}{2}} \tan{\left(1 \right)}}{3}+C$$

Answer

$$$\int \left(- \sqrt{x} \tan{\left(1 \right)}\right)\, dx = - \frac{2 x^{\frac{3}{2}} \tan{\left(1 \right)}}{3} + C$$$A

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Integral of $$$- \sqrt{x} \tan{\left(1 \right)}$$$

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