Property 5
\(\blacksquare\quad\displaystyle\int\left(\samefrac{df(x)}{dx}\right)\,dx=f(x)+c\)
\(\blacksquare\quad\samefrac d{dx}(\displaystyle\int f(x)\,dx)=f(x)\)
\(\blacksquare\quad d\displaystyle\int f(x)\,dx=f(x)\,dx\)
1.
\(\samefrac d{dx}\displaystyle\int(x^3+\sin x)\,dx=?\)
2.
\(d\displaystyle\int(\cos^2x)\,dx=?\)
3.
\(\displaystyle\int \left(\samefrac d{dx}\cdot\left(\samefrac{x^2+1}{3x}\right)\right)\,dx=?\)
4.
\(\displaystyle\int f(x)\,dx=x^3+4x+3\)
\(\Rightarrow f(x)=?\)
5.
\(\displaystyle\int f(x)\,dx=x^2-\sin x+7\)
\(\Rightarrow f(x)=?\)
6.
\(\displaystyle\int x\cdot f(x)\,dx=3x^4-x^3+10\)
\(\Rightarrow f(x)=?\)
7.
\(\samefrac{df(x)}{dx}=6x^2-10x\)
\(\Rightarrow f(x)=?\)
8.
\(f'(x)=x^3+\cos x\)
\(\Rightarrow f(x)=?\)
9.
\(f'(x)=4x+1\)
\(f(0)=7\)
\(\Rightarrow f(1)=?\)
10.
\(f'(x)=6x^2-2x+1\)
\(f(0)=2\)
\(\Rightarrow f(2)=?\)
11.
\(f'(x)=x^3-x\)
\(f(1)=7\)
\(\Rightarrow f(0)=?\)
12.
\(d\displaystyle\int(x\ln x)\,dx=?\)
13.
\(\displaystyle\int d(\sin x)=?\)