Question

Integrate the following functions wrt x
I) 1/(2x+1)^3/2
ii) sin(2x+3)
iii)cosec(4x)
iv)1/√(1-9x^2)
v)1/(1+4x)

Answer 1

*all basic integrals you can find here https://math2.org/math/integrals/tableof.htm*$\int \frac{1}{(2x+1)^{\frac{3}{2}}}dx = \int \frac{1}{2(2x+1)^{\frac{3}{2}}}d(2x+1) = \frac{1}{\sqrt{2x+1}} + C$

$\int sin(2x+3)dx = \int \frac{sin(2x+3)}{2}d(2x+3) = -cos(2x+3) + C$

$\int cosec(4x)dx = \int \frac{cosec(4x)}{4}d(4x) =-\frac{ln|cosec(4x)+cot(4x)|}{4} + C$

$\int \frac{1}{\sqrt{1-9x^2}}dx = \int \frac{1}{3\sqrt{1-9x^2}}d(3x) = \frac{sin^{-1}3x}{3} + C$

$\int \frac{1}{1+4x}dx = \int \frac{1}{4(1+4x)}d(4x) = \frac{ln(1+4x)}{4} + C$