Property 6
Concept of Differential
\(\blacksquare\quad df(x)=\left(\samefrac{df(x)}{dx}\right)\,dx\)
Example
\(1.\quad d(\sin x)=\cos x\,dx\)
\(2.\quad d(x^3)=3x^2\,dx\)
\(3.\quad d(x^2+x)=(2x+1)\,dx\)
Example
\(\displaystyle\int2x^4d(x^2)\)
Answer – 1
\(\begin{aligned}x^2&=t\\x^4&=t^2\end{aligned}\)
\(\Rightarrow\displaystyle\int2t^2\,dt=\samefrac{2t^3}3+c=\samefrac{2x^6}3+c\)
Answer – 2
\(d(x^2)=2x\cdot dx\)
\(\displaystyle\int2x^4d(x^2)\)
\(\Rightarrow\displaystyle\int2x^4\cdot2x\,dx\)
\(\Rightarrow\displaystyle\int4x^5\,dx\)
\(\Rightarrow4\samefrac{x^6}6+c=\samefrac{2x^6}3+c\)
1.
\(\displaystyle\int2x\,d(x^3)=?\)
2.
\(\displaystyle\int\sin^3x\,d(\sin x)=?\)
3.
\(\displaystyle\int(x^4+x^2)\,d(x^3)=?\)
4.
\(\displaystyle\int d(\sin x)=?\)
5.
\(\displaystyle\int d(\ln x)=?\)
6.
\(\displaystyle\int d(2^x)=?\)
7.
\(\displaystyle\int d(\tan x)=?\)
8.
\(\displaystyle\int d(e^x-e^{-x}+\sin x)=?\)
9.
\(\displaystyle\int(x^2-1)\,d(-x)=?\)
10.
\(\displaystyle\int\samefrac{d(e^{2x}+1)}{e^{2x}}=?\)
11.
\(\displaystyle\int\samefrac{d(x^3+1)}{x^3+1}=?\)
12.
\(\displaystyle\int\sin^2x\,d(\cot x)=?\)
13.
\(\samefrac d{dx}\left(\displaystyle\int\samefrac1{1+x^4}\,dx\right)=?\)