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Find the integral $\int x\cos\left(2x^2+3\right)dx$

Step-by-step Solution

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Final answer to the problem

$\frac{1}{4}\sin\left(2x^2+3\right)+C_0$
Got another answer? Verify it here!

Step-by-step Solution

How should I solve this problem?

  • Integrate using trigonometric identities
  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using basic integrals
  • Product of Binomials with Common Term
  • FOIL Method
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Can't find a method? Tell us so we can add it.
1

We can solve the integral $\int x\cos\left(2x^2+3\right)dx$ 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 $2x^2+3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=2x^2+3$
2

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

$du=4xdx$
3

Isolate $dx$ in the previous equation

$\frac{du}{4x}=dx$
4

Substituting $u$ and $dx$ in the integral and simplify

$\int\frac{\cos\left(u\right)}{4}du$
5

Take the constant $\frac{1}{4}$ out of the integral

$\frac{1}{4}\int\cos\left(u\right)du$
6

Divide $1$ by $4$

$\frac{1}{4}\int\cos\left(u\right)du$
7

Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$

$\frac{1}{4}\sin\left(u\right)$
8

Replace $u$ with the value that we assigned to it in the beginning: $2x^2+3$

$\frac{1}{4}\sin\left(2x^2+3\right)$
9

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{4}\sin\left(2x^2+3\right)+C_0$

Final answer to the problem

$\frac{1}{4}\sin\left(2x^2+3\right)+C_0$

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Function Plot

Plotting: $\frac{1}{4}\sin\left(2x^2+3\right)+C_0$

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4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Main Topic: Integral Calculus

Integration assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Used Formulas

See formulas (2)

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