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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 $u=2x^2+3$ Intermediate steps 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 $\frac{du}{4x}=dx$ Intermediate steps 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 $\int\frac{\cos\left(u\right)}{4}du$ Intermediate steps Take the constant $\frac{1}{4}$ out of the integral $\frac{1}{4}\int\cos\left(u\right)du$ Divide $1$ by $4$ $\frac{1}{4}\int\cos\left(u\right)du$ 6 Take the constant $\frac{1}{4}$ out of the integral $\frac{1}{4}\int\cos\left(u\right)du$ 7 Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$ $\frac{1}{4}\sin\left(u\right)$ Intermediate steps $\frac{1}{4}\sin\left(2x^2+3\right)$ 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$\frac{1}{4}\sin\left(2x^2+3\right)+C_0$
Solve Equations with SubstitutionSubstitution is the process of replacing a variable in an expression with its actual value. If you are given an equation like \(4z + 6 = x + z\), told that \(z = 2\), and asked to solve for x, what do you do? The first step is to substitute 2 for every z in the problem: $$ 4z+6=x+z $$ $$ \cancel{4z}4*2+6=x+\cancel{z}2 $$ This leaves us a more manageable equation, \(4 * 2 + 6 = x + 2\). After simplifying things a little we get \(14 = x + 2\). Subtract 2 from each side and you'll discover that \(x = 12\). Substitution can also involve more complicated problems, like \(5x^2 + 2y -6 = z\), where \(y = -1\) and \(x = 4\). Just substitute in the same way, placing the value of each variable in place of the letter and simplifying. $$ 5x^2+2y-6=z $$ $$ 5*4^2+2(-1)-6=z $$ $$ 5*16-2-6=z $$ $$ 72=z $$ Those example are pretty straightforward. You not even have needed that explanation, but it leads to more complex examples. The idea was first to understand the concept of substitution before we go further. There is another form of substitution where we insert a variable to stand for a particular expression. In some cases you will be dealing with incredibly complex problems, and it may be easier to substitute a variable, like y for part of the problem. Then, when you have solved the value of y, you can plug back in the original expression.
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What are the 5 steps in solving equations by substitution?Steps to Solving by Substitution:. Step One→ Solve one equation for either x or y.. Step Two→ Substitute the expression from step one into the 2nd equation.. Step Three→ Solve the second equation for the given variable.. Step Four→ Plug you solution back into the first equation.. Step Five→ Write your solution as a point.. How do you solve by substitution grade 10?To solve using substitution, follow these four steps: Step 1: Isolate a variable. Step 2: Plug the result of Step 1 into the other equation and solve for one variable. Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.
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