Answer with Step-by-step explanation:
We are given that C=0
We have to find the anti-derivative of each function
[tex]a.h(x)=sec^2 x[/tex]
We know that [tex] \int sec^2 xdx=tanx+C[/tex]
Apply this formula then, we get
[tex]\int sec^2x dx=tanx +C[/tex]
Substitute C=0
Then, we get [tex]\int h(x) dx=tan x[/tex]
Verification :
Differentiate w.r.t x
Then, we get
[tex]h(x)=sec^2 x[/tex]
[tex]\frac{d tanx}{dx}=sec^2 x[/tex]
[tex]b.g(x)=\frac{8}{9} sec^2 \frac{x}{9}[/tex]
[tex]\int g(x) dx=\frac{8}{9}\int sec^2\frac{x}{9} dx=\frac{8}{9}\times 9tan\frac{x}{9}+C[/tex]
Substitute C=0
[tex]\int g(x) dx=8 tan\frac{x}{9}[/tex]
Verification:
Differentiate w.r.t x
[tex]g(x)=8\times \frac{1}{9} sec^2\frac{x}{9}=\frac{8}{9} sec^2\frac{x}{9}[/tex]
[tex]c.k(x)=sec^2\frac{9x}{8}[/tex]
[tex]\int k(x) dx=\int \sec^28}{9}tan\frac{9x}{8}+C[/tex]
Substitute C=0
[tex]\int k(x) dx=\frac{8}{9} tan\frac{9x}{8}[/tex]
Verification: Differentiate w.r.t x
[tex]k(x)=\frac{8}{9}\times \frac{9}{8} sec^2\frac{9x}{8}=sec^2\frac{9x}{8}[/tex]
