Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Choose an option
- Integrate by partial fractions
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
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We can solve the integral $\int\frac{1}{\sin\left(x\right)-1}dx$ by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of $t$ by setting the substitution
Hence
Substituting in the original integral we get
Combine $\frac{2t}{1+t^{2}}-1$ in a single fraction
Multiplying fractions $\frac{1}{\frac{2t-\left(1+t^{2}\right)}{1+t^{2}}} \times \frac{2}{1+t^{2}}$
Divide fractions $\frac{2}{\frac{2t-\left(1+t^{2}\right)}{1+t^{2}}\left(1+t^{2}\right)}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Simplify the fraction $\frac{2\left(1+t^{2}\right)}{\left(2t-\left(1+t^{2}\right)\right)\left(1+t^{2}\right)}$ by $1+t^{2}$
Simplifying
Solve the product $-\left(1+t^{2}\right)$
The trinomial $2t+1+1t^{2}$ is a perfect square trinomial, because it's discriminant is equal to zero
Using the perfect square trinomial formula
Factoring the perfect square trinomial
Rewrite the expression $\frac{2}{2t-\left(1+t^{2}\right)}$ inside the integral in factored form
Simplify the division $2$ by $-1$
We can solve the integral $\int\frac{-2}{\left(t-1\right)^{2}}dt$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $t-1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Differentiate both sides of the equation $u=t-1$
Find the derivative
The derivative of a sum of two or more functions is the sum of the derivatives of each function
Now, in order to rewrite $dt$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Substituting $u$ and $dt$ in the integral and simplify
Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$
The integral of a function times a constant ($-2$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$
Simplify the fraction $-2\left(\frac{u^{-1}}{-1}\right)$
Replace $u$ with the value that we assigned to it in the beginning: $t-1$
Replace $u$ with the value that we assigned to it in the beginning: $t-1$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Replace $t$ with the value that we assigned to it in the beginning: $\tan\left(\frac{x}{2}\right)$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
