Step-by-step Solution

Integrate $x\cos\left(2x^2+3\right)$ with respect to x

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Final Answer

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

Step-by-step explanation

Problem to solve:

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

Choose 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

$u=2x^2+3$
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

$du=4xdx$
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Isolate $dx$ in the previous equation

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

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

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

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

Final Answer

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

Main topic:

Calculus

Related formulas:

1. See formulas

Time to solve it:

~ 0.03 s (SnapXam)