Question

Evaluate:

1)$\int \frac{x^3+3x+4}{\sqrt{x}}dx$

2)$\int \sqrt{x}(3x^2+2x+3)dx$

3)$\int (2x^2-3\sin x+5\sqrt{x})dx$

4) $\int \frac{{\sec }^{2}x}{{\csc }^{2}x}dx$

Answer 1

$\left(I\right)$

$\int \frac{x^3+3x+4}{\sqrt{x}}dx$

$\Rightarrow \int (x^{3-1/2}+{3x}^{1-1/2}+4x^{-1/2})dx$

$\Rightarrow \int x^{5/2}+3x^{1/2}+{4x}^{-1/2}$

$\Rightarrow \frac{x^{7/2}}{7/2}+\frac{x^{3/2}}{3/2}+\frac{8}{x^{1/2}}+c$

$\left(II\right)$

$\int \sqrt{x}({3x}^2+2x+3)dx$

$\Rightarrow \int ({3x}^{5/2}+{2x}^{3/2}+3x^{1/2})dx$

$\Rightarrow \frac{{3x}^{7/2}}{7/2}+\frac{{2x}^{5/2}}{5/2}+\frac{{3x}^{3/2}}{3/2}+c$

$\left(III\right)$

$\int ({2x}^2-3sinx+5\sqrt{x})dx$

$\Rightarrow \frac{{2x}^3}{3}+3\cos x+\frac{{5x}^{3/2}}{3/2}+c$

$\left(IV\right)$

$\int \frac{{\sec }^{2}x}{{cosec}^{2}x}dx$

$=\int {tan}^{2}xdx$

$\Rightarrow \int ({\sec }^{2}x+1)dx$

$\Rightarrow \tan x-x+c$