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- 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 $\int\frac{3\sin\left(x\right)+4\cos\left(x\right)}{4\sin\left(x\right)-3\cos\left(x\right)}dx$ 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 $4\sin\left(x\right)-3\cos\left(x\right)$ 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 $u=4\sin\left(x\right)-3\cos\left(x\right)$
Find the derivative
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$
The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$
Now, in order to rewrite $dx$ 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
Isolate $dx$ in the previous equation
Simplify the fraction $\frac{\frac{3\sin\left(x\right)+4\cos\left(x\right)}{u}}{4\cos\left(x\right)+3\sin\left(x\right)}$ by $4\cos\left(x\right)+3\sin\left(x\right)$
Substituting $u$ and $dx$ in the integral and simplify
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Replace $u$ with the value that we assigned to it in the beginning: $4\sin\left(x\right)-3\cos\left(x\right)$
Replace $u$ with the value that we assigned to it in the beginning: $4\sin\left(x\right)-3\cos\left(x\right)$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
