Solve \int_{-\infty}^{\infty}x^{-3}dx

Evaluate

\int_{- (+ \infty)}^{+ \infty} x^{- 3} d x

Evaluate

\int_{- \infty}^{+ \infty} x^{- 3} d x

Use function domain and discontinuity points to transform the expression with the formula \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

\int_{- \infty}^{- 1} x^{- 3} d x + \int_{- 1}^{0} x^{- 3} d x + \int_{0}^{1} x^{- 3} d x + \int_{1}^{+ \infty} x^{- 3} d x

By definition,rewrite the improper integral using one-sided limit and a definite integral

a \to - \infty lim (\int_{a}^{- 1} x^{- 3} d x) + \int_{- 1}^{0} x^{- 3} d x + \int_{0}^{1} x^{- 3} d x + \int_{1}^{+ \infty} x^{- 3} d x

By definition,rewrite the improper integral using one-sided limit and a definite integral

a \to - \infty lim (\int_{a}^{- 1} x^{- 3} d x) + a \to 0^{-} lim (\int_{- 1}^{a} x^{- 3} d x) + \int_{0}^{1} x^{- 3} d x + \int_{1}^{+ \infty} x^{- 3} d x

By definition,rewrite the improper integral using one-sided limit and a definite integral

a \to - \infty lim (\int_{a}^{- 1} x^{- 3} d x) + a \to 0^{-} lim (\int_{- 1}^{a} x^{- 3} d x) + a \to 0^{+} lim (\int_{a}^{1} x^{- 3} d x) + \int_{1}^{+ \infty} x^{- 3} d x

By definition,rewrite the improper integral using one-sided limit and a definite integral

a \to - \infty lim (\int_{a}^{- 1} x^{- 3} d x) + a \to 0^{-} lim (\int_{- 1}^{a} x^{- 3} d x) + a \to 0^{+} lim (\int_{a}^{1} x^{- 3} d x) + a \to + \infty lim (\int_{1}^{a} x^{- 3} d x)

Evaluate the integral

More Steps

a \to - \infty lim (- \frac{1}{2} + \frac{1}{2 a ^{2}}) + a \to 0^{-} lim (\int_{- 1}^{a} x^{- 3} d x) + a \to 0^{+} lim (\int_{a}^{1} x^{- 3} d x) + a \to + \infty lim (\int_{1}^{a} x^{- 3} d x)

Evaluate the integral

More Steps

a \to - \infty lim (- \frac{1}{2} + \frac{1}{2 a ^{2}}) + a \to 0^{-} lim (- \frac{1}{2 a ^{2}} + \frac{1}{2}) + a \to 0^{+} lim (\int_{a}^{1} x^{- 3} d x) + a \to + \infty lim (\int_{1}^{a} x^{- 3} d x)

Evaluate the integral

More Steps

a \to - \infty lim (- \frac{1}{2} + \frac{1}{2 a ^{2}}) + a \to 0^{-} lim (- \frac{1}{2 a ^{2}} + \frac{1}{2}) + a \to 0^{+} lim (- \frac{1}{2} + \frac{1}{2 a ^{2}}) + a \to + \infty lim (\int_{1}^{a} x^{- 3} d x)

Evaluate the integral

More Steps

a \to - \infty lim (- \frac{1}{2} + \frac{1}{2 a ^{2}}) + a \to 0^{-} lim (- \frac{1}{2 a ^{2}} + \frac{1}{2}) + a \to 0^{+} lim (- \frac{1}{2} + \frac{1}{2 a ^{2}}) + a \to + \infty lim (- \frac{1}{2 a ^{2}} + \frac{1}{2})

Evaluate the limit

More Steps

- \frac{1}{2} + a \to 0^{-} lim (- \frac{1}{2 a ^{2}} + \frac{1}{2}) + a \to 0^{+} lim (- \frac{1}{2} + \frac{1}{2 a ^{2}}) + a \to + \infty lim (- \frac{1}{2 a ^{2}} + \frac{1}{2})

Evaluate the limit

More Steps

- \frac{1}{2} + (- \infty) + a \to 0^{+} lim (- \frac{1}{2} + \frac{1}{2 a ^{2}}) + a \to + \infty lim (- \frac{1}{2 a ^{2}} + \frac{1}{2})

Evaluate the limit

More Steps

- \frac{1}{2} + (- \infty) + (+ \infty) + a \to + \infty lim (- \frac{1}{2 a ^{2}} + \frac{1}{2})

Evaluate the limit

More Steps

- \frac{1}{2} + (- \infty) + (+ \infty) + \frac{1}{2}

Solution

Diverges