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Integration Techniques Calculator

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1

Solved example of integration by substitution

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

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 $f'(x) = nx^{n-1}$

$4x$
3

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

$du=4xdx$

4

Isolate $dx$ in the previous equation

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

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

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

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

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

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

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

Divide $1$ by $4$

$\frac{1}{4}\int\cos\left(u\right)du$
8

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

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

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

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

$\frac{1}{4}\sin\left(2x^2+3\right)$
10

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

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

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