Solve ∫ (from - 1 to 1) of (2x^2*x^3) wrt x

$$\int _ { - 1 } ^ { 1 } ( 2 x ^ { 2 } \cdot x ^ { 3 } ) d x$$

Evaluate

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Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\int _{-1}^{1}2x^{5}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 2x^{5}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$2\int x^{5}\mathrm{d}x$$

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}.

$$\frac{x^{6}}{3}$$

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.

$$\frac{1^{6}}{3}-\frac{\left(-1\right)^{6}}{3}$$

Simplify.

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