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Kalkylator för bestämda och oegentliga integraler
Beräkna bestämda och oegentliga integraler steg för steg
Kalkylatorn försöker beräkna bestämda integraler (dvs. med integrationsgränser), inklusive oegentliga, och visar stegen.
Solution
Your input: calculate $$$\int_{3}^{c}\left( c + f^{2} x^{2} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\left(c + f^{2} x^{2}\right)d x}=x \left(c + \frac{f^{2} x^{2}}{3}\right)$$$ (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.
$$$\left(x \left(c + \frac{f^{2} x^{2}}{3}\right)\right)|_{\left(x=c\right)}=c \left(\frac{c^{2} f^{2}}{3} + c\right)$$$
$$$\left(x \left(c + \frac{f^{2} x^{2}}{3}\right)\right)|_{\left(x=3\right)}=3 c + 9 f^{2}$$$
$$$\int_{3}^{c}\left( c + f^{2} x^{2} \right)dx=\left(x \left(c + \frac{f^{2} x^{2}}{3}\right)\right)|_{\left(x=c\right)}-\left(x \left(c + \frac{f^{2} x^{2}}{3}\right)\right)|_{\left(x=3\right)}=c \left(\frac{c^{2} f^{2}}{3} + c\right) - 3 c - 9 f^{2}$$$
Answer: $$$\int_{3}^{c}\left( c + f^{2} x^{2} \right)dx=c \left(\frac{c^{2} f^{2}}{3} + c\right) - 3 c - 9 f^{2}$$$