Question

найти первообразную,график которой проходит через точку^
1. f(X)=8x^3-3x^2+4x-2 A(-1 , 2)
2.f(x)=3cos3x A( п\18, 1)

Answer 1

$\displaystyle 1)\quad f(x)=8x^3-3x^2+4x-2\\\\\text{F}(x)=\int\limits {(8x^3-3x^2+4x-2)} \, dx =\frac{8x^4}4-\frac{3x^3}3+\frac{4x^2}2-2x+\text{C}=\\\\=2x^4-x^3+2x^2-2x+\text{C}, \quad \text{A}(-1,2)\\\\2=2\cdot(-1)^4-(-1)^3+2\cdot(-1)^2-2\cdot(-1)+\text{C}\\\\2=2+1+2+2+\text{C}\\\\\text{C}+5=0\\\\\text{C}=-5\\\\\boxed{\text{F}(x)=2x^4-x^3+2x^2-2x-5}$

$\displaystyle 2)\quad f(x)=3\cos(3x)\\\\\text{F}(x)=\int\limits {3\cos(3x)} \, dx =\int\limits {\cos(3x)} \, d(3x)=\sin(3x)+\text{C}, \quad A\bigg(\frac{\pi}{18},\, 1\bigg)\\\\1=\sin\bigg(3\cdot\frac{\pi}{18}\bigg)+\text{C}\\\\\text{C}=1-\sin\bigg(\frac{\pi}6\bigg)=1-\frac{1}2=\frac{1}2\\\\\boxed{\text{F}(x)=\sin(3x)+\frac{1}2}$