Answer:
-2x cos x + 2 sin x + C
Step-by-step explanation:
∫ (2x sin x) dx
If u = 2x, then du = 2 dx.
If dv = sin dx, then v = -cos x.
∫ u dv = uv − ∫ v du
= 2x (-cos x) − ∫ (-cos x) (2 dx)
= -2x cos x + 2 ∫ cos x dx
= -2x cos x + 2 sin x + C
Answer:
[tex]\displaystyle \large{2 \sin x - 2x \cos x + C}[/tex]
Step-by-step explanation:
We are given the indefinite integral:
[tex]\displaystyle \large{\int {2x \sin x} \ dx }[/tex]
Using by-part method, we have to substitute u-term and dv appropriately.
By-part is an integration of product rules, when integrated the product rules of differentiation, we’ll obtain:
[tex]\displaystyle \large{\int {u} \ dv = uv - \int {v} \ du}[/tex]
Above is by-part method/formula.
Where 4 terms are presented:
Our main terms to substitute are u and dv which mean u-term has to be a function that’s differentiatable and dv has to be a function that’s integratable.
The main concept of by-part is to understand how to substitute appropriately which you can simply follow below:
LIATE
Stands for Logarithm, Inverse (Trigonometry), Algebraic, Trigonometric and Exponential.
These are in orders from first to last on what to let u-term first. That means logarithm functions must be the first to substitute themselves as u-term, so if you encounter a logarithmic function and a polynomial function, you must let u = logarithmic function while dv = polynomial.
In this case, we have 2x which is polynomial and sin(x) which is trigonometric. According to LIATE, we have to let Algebraic or Polynomial 2x be first to substitute as u-term, that means our dv is trigonometric sin(x).
Therefore, we have:
Now, substitute these terms in accordingly to formula of by-part.
[tex]\displaystyle \large{\int {2x \sin x} \ dx = 2x \cdot (-\cos x) - \int {-\cos x \cdot 2 \ dx}}\\ \displaystyle \large{\int {2x \sin x} \ dx = -2x \cos x + \int {2 \cos x \ dx}}\\ \displaystyle \large{\int {2x \sin x} \ dx = -2x \cos x + 2 \int {\cos x \ dx}}\\ \displaystyle \large{\int {2x \sin x} \ dx = -2x \cos x + 2 \cdot \sin x + C}\\ \displaystyle \large{\int {2x \sin x} \ dx = -2x \cos x + 2 \sin x + C}[/tex]
__________________________________________________
Summary
Property
[tex]\displaystyle \large{ \int {k f(x)} \ dx = k \int{f(x)} \ dx \ \ \ \ \tt{(k \ \ \ is \ \ \ a \ \ \ constant.)}\\[/tex]
Only shown in explanation.
By-Part
[tex]\displaystyle \large{\int u \ dv = uv- \int v \ du}[/tex]
LIATE
The functions in order that should be u-term from first to last.
These functions above do not have integration formula by default.
Indefinite Integral
Make sure to always add + C after evaluating the integral, regardless what multiplies or attempts to affect + C, we must always add + C.
__________________________________________________
