Practice section 1 of 10

Differentiate $(5x^2+2x+1)^8$ with respect to $x$. Hence find $\int(5x+1)(5x^2+2x+1)^7\,dx$.

Integrate with respect to $x$: (a) $\sqrt{x}(1-x)$; (b) $(x^{1/2}+3)^2$; (c) $(5x+4)^5$; (d) $x(2x^2-3)^4$; (e) $\frac{2x+3}{(x^2+3x)^3}$.

Integrate with respect to $x$: (a) $\frac3{x^2}+6x^2$; (b) $\frac{x^3-x^2+4x-7}{\sqrt{x}}$.

Integrate with respect to $x$: (a) $\frac1{2x+7}$; (b) $\frac{x^2+2}{x^3+6x}$; (c) $\frac{x^2}{x+1}$. How do we handle integrals containing improper rational functions?

Integrate with respect to $x$: (a) $3x^2+\frac1x$; (b) $\frac1{2-3x}$; (c) $\frac{x}{1+x^2}$.

Integrate with respect to $x$: (a) $2e^{2x-1}$; (b) $3xe^{x^2-1}$; (c) $\sqrt{e^x}-\frac1{\sqrt{e^x}}$; (d) $\frac{e^{\sin^{-1}x}}{\sqrt{1-x^2}}$; (e) $x^3 5^{1-x^4}$.