Property 8
\(\displaystyle\int f(x)\,dx=G(x)+c\)
\(\Rightarrow\displaystyle\int f(ax+b)\,dx=\samefrac{G(ax+b)}a+c\)
1.
\(\displaystyle\int (2x+7)^4\,dx=?\)
2.
\(\displaystyle\int (3x+1)^2\,dx=?\)
3.
\(\displaystyle\int \sin(4x+1)\,dx=?\)
4.
\(\displaystyle\int \cos(5x-7)\,dx=?\)
5.
\(\displaystyle\int e^{3x-1}\,dx=?\)
6.
\(\displaystyle\int \samefrac1{7x-1}\,dx=?\)
7.
\(\displaystyle\int \sqrt{5x-1}\,dx=?\)
8.
\(\displaystyle\int \sqrt[3]{2x+1}\,dx=?\)
Property 9
\(\blacksquare\quad\displaystyle\int\sin^m x\cdot\cos^n x\,dx\)
An expression with an even exponent is called u.
\(\blacksquare\quad\displaystyle\int\cos^2x\,dx=\displaystyle\int\samefrac{\cos(2x)+1}2\,dx\)
\(\blacksquare\quad\displaystyle\int\sin^2x\,dx=\displaystyle\int\samefrac{1-\cos(2x)}2\,dx\)
1.
\(\displaystyle\int \sin x\cdot\cos x\,dx=?\)
2.
\(\displaystyle\int \sin^4x\cdot\cos x\,dx=?\)
3.
\(\displaystyle\int \cos^5x\cdot\sin x\,dx=?\)
4.
\(\displaystyle\int \sin^2x\cdot\cos^3x\,dx=?\)
5.
\(\displaystyle\int \sin^3x\,dx=?\)
6.
\(\displaystyle\int \cos^3x\,dx=?\)
7.
\(\displaystyle\int \sin^2x\,dx=?\)
8.
\(\displaystyle\int \cos^2x\,dx=?\)