정적분 및 가적분 계산기

이 계산기는 단계별 풀이를 보여 주면서 상한과 하한이 있는 정적분(진정적분 포함)을 계산하려고 시도합니다.

Enter a function:

Integrate with respect to:

Enter a lower limit:

If you need `-oo`, type -inf.

Enter an upper limit:

If you need `oo`, type inf.

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

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Solution

Your input: calculate $$$\int_{\frac{1}{2}}^{1}\left( \sin{\left(x \right)} - \frac{1}{x} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\left(\sin{\left(x \right)} - \frac{1}{x}\right)d x}=- \ln{\left(\left|{x}\right| \right)} - \cos{\left(x \right)}$$$ (for steps, see indefinite integral calculator)

According to the Fundamental Theorem of Calculus, $$$\int_a^b F(x) dx=f(b)-f(a)$$$, so just evaluate the integral at the endpoints, and that's the answer.

$$$\left(- \ln{\left(\left|{x}\right| \right)} - \cos{\left(x \right)}\right)|_{\left(x=1\right)}=- \cos{\left(1 \right)}$$$

$$$\left(- \ln{\left(\left|{x}\right| \right)} - \cos{\left(x \right)}\right)|_{\left(x=\frac{1}{2}\right)}=- \cos{\left(\frac{1}{2} \right)} + \ln{\left(2 \right)}$$$

$$$\int_{\frac{1}{2}}^{1}\left( \sin{\left(x \right)} - \frac{1}{x} \right)dx=\left(- \ln{\left(\left|{x}\right| \right)} - \cos{\left(x \right)}\right)|_{\left(x=1\right)}-\left(- \ln{\left(\left|{x}\right| \right)} - \cos{\left(x \right)}\right)|_{\left(x=\frac{1}{2}\right)}=- \ln{\left(2 \right)} - \cos{\left(1 \right)} + \cos{\left(\frac{1}{2} \right)}$$$

Answer: $$$\int_{\frac{1}{2}}^{1}\left( \sin{\left(x \right)} - \frac{1}{x} \right)dx=- \ln{\left(2 \right)} - \cos{\left(1 \right)} + \cos{\left(\frac{1}{2} \right)}\approx -0.355866924537712$$$

Please try a new game Rotatly

정적분 및 가적분 계산기

정적분과 광의적분을 단계별로 계산하세요

Solution