Makayla Boyd

2022-06-25

Integral $\int \frac{dx}{{\mathrm{sin}}^{2}(x)+\mathrm{sin}(2x)}$

kejohananws

Beginner2022-06-26Added 19 answers

$\begin{array}{rl}\int \frac{dx}{{\mathrm{sin}}^{2}(x)+\mathrm{sin}(2x)}& =\int \frac{{\mathrm{sec}}^{2}(x)}{{\mathrm{sin}}^{2}(x){\mathrm{sec}}^{2}(x)+2\mathrm{sin}(x)\mathrm{cos}(x){\mathrm{sec}}^{2}(x)}dx\\ & =\int \frac{{\mathrm{sec}}^{2}(x)}{{\mathrm{tan}}^{2}(x)+2\mathrm{tan}(x)}dx\end{array}$

Letting $u:=\mathrm{tan}(x)$, then $du={\mathrm{sec}}^{2}(x)dx$ gives

$\begin{array}{rl}\int \frac{dx}{{\mathrm{sin}}^{2}(x)+\mathrm{sin}(2x)}& =\int \frac{du}{{u}^{2}+2u}\\ & =\int \frac{du}{(u+1{)}^{2}-1}\end{array}$

Letting z:=u+1, then dz=du gives

$\begin{array}{rl}\int \frac{dx}{{\mathrm{sin}}^{2}(x)+\mathrm{sin}(2x)}& =\int \frac{dz}{{z}^{2}-1}\\ & =-\text{artanh}(z)+C\\ & =-\text{artanh}(u+1)+C\end{array}$

Reersing the final substitution gives

$\int \frac{dx}{{\mathrm{sin}}^{2}(x)+\mathrm{sin}(2x)}=-\text{artanh}(\mathrm{tan}(x)+1)+C$

Letting $u:=\mathrm{tan}(x)$, then $du={\mathrm{sec}}^{2}(x)dx$ gives

$\begin{array}{rl}\int \frac{dx}{{\mathrm{sin}}^{2}(x)+\mathrm{sin}(2x)}& =\int \frac{du}{{u}^{2}+2u}\\ & =\int \frac{du}{(u+1{)}^{2}-1}\end{array}$

Letting z:=u+1, then dz=du gives

$\begin{array}{rl}\int \frac{dx}{{\mathrm{sin}}^{2}(x)+\mathrm{sin}(2x)}& =\int \frac{dz}{{z}^{2}-1}\\ & =-\text{artanh}(z)+C\\ & =-\text{artanh}(u+1)+C\end{array}$

Reersing the final substitution gives

$\int \frac{dx}{{\mathrm{sin}}^{2}(x)+\mathrm{sin}(2x)}=-\text{artanh}(\mathrm{tan}(x)+1)+C$

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.