Question

Please help !!! I’ll mark brainliest

Answer 1

A) State the chain rule for integration

Ans. The chain rule for integration is also known as "  Integration by substitution "

Integration by substitution is taken in order to make integration solve easily in few steps.

For, $I = \int\limits (x+2)^{2} \,dx$

Instead of expanding term $(x+2)^{2}$

With substitution of  $u = (x+2)$ and $du= 1 dx$

We simplified the integration as

$I = \int\limits (u)^{2} \, du$

$I = \frac{(u)^{3}}{3}+C$

By replacaing value of u=x+2

$I = \frac{(x+2)^{3}}{3}+C$

B) State the rule of differentiation for the sine function.

Ans. We know that $\frac{d}{dx}Sinx dx = Cosx$

C) Find the indefinite integral using substitution.

Ans.

Given, $I = \int\limits {\frac{Cos14x}{Sin14x} } \, dx$

Take y = Sin14x

Differentiating both side

$dy=14Cos14x dx$

$\frac{dy}{14} = Cos14x\, dx$

Substituting values in integration,

$I = \int\limits {\frac{Cos14x}{Sin14x} } \, dx$

$I = \int\limits {\frac{1}{y} } \,\frac{dy}{14}$

$I = \frac{1}{14}\int\limits {\frac{1}{y} } \,dy$

$I = \frac{1}{14} lny + C$

Replacing values in the integration

$I = \frac{1}{14} ln(14Sin14x) + C$

D)Check your work by taking a derivative of your answer from part C.

Ans.

Answer for Part C is $I = \frac{1}{14} ln(14Sin14x) + C$

Differentiating the answer

we get,

$=\frac{1}{14}\frac{d}{dx}[ ln(Sin14x) + C]\\=\frac{1}{14}\frac{1}{Sin14x} \frac{d}{dx}(Sin14x)+ \frac{d}{dx}C\\=\frac{1}{14}\frac{1}{Cos14x}(14Cos14x)\\=\frac{Cos14x}{Sin14x} \\ =I$