Integrate x^3+3x^2−3​

\int\left(x^3+3x^2\cdot x​\right)dx+\int\left(x^3+3x^{2−3​}\right)dx

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Answer

$3\frac{x^{\left(1+2−3​\right)}}{1+2−3​}+\int x​\cdot 3x^2dx+\frac{1}{2}x^{4}$

Step by step solution

Problem

$\int\left(x^3+3x^2\cdot x​\right)dx+\int\left(x^3+3x^{2−3​}\right)dx$
1

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int3x^{2−3​}dx+\int x^3dx+\int x​\cdot 3x^2dx+\int x^3dx$
2

Adding $\int x^3dx$ and $\int x^3dx$

$\int3x^{2−3​}dx+\int x​\cdot 3x^2dx+2\int x^3dx$
3

Taking the constant out of the integral

$3\int x^{2−3​}dx+\int x​\cdot 3x^2dx+2\int x^3dx$
4

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$3\frac{x^{\left(1+2−3​\right)}}{1+2−3​}+\int x​\cdot 3x^2dx+2\int x^3dx$
5

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$3\frac{x^{\left(1+2−3​\right)}}{1+2−3​}+\int x​\cdot 3x^2dx+2\frac{x^{4}}{4}$
6

Simplify the fraction

$3\frac{x^{\left(1+2−3​\right)}}{1+2−3​}+\int x​\cdot 3x^2dx+\frac{1}{2}x^{4}$

Answer

$3\frac{x^{\left(1+2−3​\right)}}{1+2−3​}+\int x​\cdot 3x^2dx+\frac{1}{2}x^{4}$

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Problem Analysis

Main topic:

Integral calculus

Time to solve it:

0.22 seconds

Views:

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