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

## Step-by-step Solution

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###  Videos

$\frac{1}{4}\sin\left(2x^2+3\right)+C_0$
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##  Step-by-step Solution 

Problem to solve:

$\int\left(x\cdot\cos\left(2x^2+3\right)\right)dx$

Specify the solving method

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

$u=2x^2+3$

Differentiate both sides of the equation $u=2x^2+3$

$du=\frac{d}{dx}\left(2x^2+3\right)$

Find the derivative

$\frac{d}{dx}\left(2x^2+3\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(2x^2\right)+\frac{d}{dx}\left(3\right)$

The derivative of the constant function ($3$) is equal to zero

$\frac{d}{dx}\left(2x^2\right)$

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

$2\frac{d}{dx}\left(x^2\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$4x$
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

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

Simplify the fraction $\frac{x\cos\left(u\right)}{4x}$ by $x$

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

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

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

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

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

Divide $1$ by $4$

$\frac{1}{4}\int\cos\left(u\right)du$
5

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

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

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

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

$\frac{1}{4}\sin\left(2x^2+3\right)$
7

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

$\frac{1}{4}\sin\left(2x^2+3\right)$
8

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

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

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

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve int(xcos(2x^2+3))dx using basic integralsSolve int(xcos(2x^2+3))dx using u-substitutionSolve int(xcos(2x^2+3))dx using integration by partsSolve int(xcos(2x^2+3))dx using tabular integration
SnapXam A2

### beta Got a different answer? Verify it!

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0
a
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m
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u
v
w
x
y
z
.
(◻)
+
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×
◻/◻
/
÷
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

$\int\left(x\cdot\cos\left(2x^2+3\right)\right)dx$

### Main topic:

Integral Calculus

~ 0.03 s