Here, we show you a step-by-step solved example of calculus. This solution was automatically generated by our smart calculator:
The cube of a binomial (sum) is equal to the cube of the first term, plus three times the square of the first by the second, plus three times the first by the square of the second, plus the cube of the second term. In other words: $(a+b)^3=a^3+3a^2b+3ab^2+b^3 = (2x)^3+3(2x)^2(3)+3(2x)(3)^2+(3)^3 =$
The power of a product is equal to the product of it's factors raised to the same power
Multiply $9$ times $4$
Multiply the single term $5x$ by each term of the polynomial $\left(8x^3+36x^2+54x+27\right)$
When multiplying exponents with same base you can add the exponents: $40x^3x$
When multiplying two powers that have the same base ($x$), you can add the exponents
When multiplying exponents with same base you can add the exponents: $180x^2x$
Rewrite the integrand $5\left(2x+3\right)^3x$ in expanded form
Expand the integral $\int\left(40x^{4}+180x^{3}+270x^2+135x\right)dx$ into $4$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of a function times a constant ($40$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $4$
Simplify the fraction $40\left(\frac{x^{5}}{5}\right)$
The integral $\int40x^{4}dx$ results in: $8x^{5}$
The integral of a function times a constant ($180$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $3$
Simplify the fraction $180\left(\frac{x^{4}}{4}\right)$
The integral $\int180x^{3}dx$ results in: $45x^{4}$
The integral of a function times a constant ($270$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$
Simplify the fraction $270\left(\frac{x^{3}}{3}\right)$
The integral $\int270x^2dx$ results in: $90x^{3}$
The integral of a function times a constant ($135$) is equal to the constant times the integral of the function
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$
Multiply the fraction and term in $135\cdot \left(\frac{1}{2}\right)x^2$
The integral $\int135xdx$ results in: $\frac{135}{2}x^2$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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