Evaluate
\frac{721}{9}\approx 80.111111111
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\int _{1}^{2}x^{2}\left(\left(x^{3}\right)^{2}+2x^{3}+1\right)\mathrm{d}x
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{3}+1\right)^{2}.
\int _{1}^{2}x^{2}\left(x^{6}+2x^{3}+1\right)\mathrm{d}x
To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.
\int _{1}^{2}x^{8}+2x^{5}+x^{2}\mathrm{d}x
Use the distributive property to multiply x^{2} by x^{6}+2x^{3}+1.
\int x^{8}+2x^{5}+x^{2}\mathrm{d}x
Evaluate the indefinite integral first.
\int x^{8}\mathrm{d}x+\int 2x^{5}\mathrm{d}x+\int x^{2}\mathrm{d}x
Integrate the sum term by term.
\int x^{8}\mathrm{d}x+2\int x^{5}\mathrm{d}x+\int x^{2}\mathrm{d}x
Factor out the constant in each of the terms.
\frac{x^{9}}{9}+2\int x^{5}\mathrm{d}x+\int x^{2}\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{8}\mathrm{d}x with \frac{x^{9}}{9}.
\frac{x^{9}}{9}+\frac{x^{6}}{3}+\int x^{2}\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}. Multiply 2 times \frac{x^{6}}{6}.
\frac{x^{9}}{9}+\frac{x^{6}}{3}+\frac{x^{3}}{3}
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.
\frac{2^{3}}{3}+\frac{2^{6}}{3}+\frac{2^{9}}{9}-\left(\frac{1^{3}}{3}+\frac{1^{6}}{3}+\frac{1^{9}}{9}\right)
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.
\frac{721}{9}
Simplify.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}