Evaluate
-1
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\int _{0}^{1}-2+2x\mathrm{d}x
Use the distributive property to multiply -2 by 1-x.
\int -2+2x\mathrm{d}x
Evaluate the indefinite integral first.
\int -2\mathrm{d}x+\int 2x\mathrm{d}x
Integrate the sum term by term.
\int -2\mathrm{d}x+2\int x\mathrm{d}x
Factor out the constant in each of the terms.
-2x+2\int x\mathrm{d}x
Find the integral of -2 using the table of common integrals rule \int a\mathrm{d}x=ax.
-2x+x^{2}
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 2 times \frac{x^{2}}{2}.
-2+1^{2}-\left(-2\times 0+0^{2}\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.
-1
Simplify.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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4 \sin \theta \cos \theta = 2 \sin \theta
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y = 3x + 4
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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}