Calculus Examples

$\int_{- \infty}^{\infty} f (x) d x$

Step 1

Split the integral at $0$ and write as a sum of limits.

$lim_{t \to - \infty} \int_{t}^{0} f (x) d x + lim_{t \to \infty} \int_{0}^{t} f (x) d x$

Step 2

Apply the constant rule.

$lim_{t \to - \infty} f (x) x]_{t}^{0} + lim_{t \to \infty} \int_{0}^{t} f (x) d x$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $f (x) x$ at $0$ and at $t$ .

$lim_{t \to - \infty} (f (x) \cdot 0) - f (x) t + lim_{t \to \infty} \int_{0}^{t} f (x) d x$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $f (x)$ by $0$ .

$lim_{t \to - \infty} 0 - f (x) t + lim_{t \to \infty} \int_{0}^{t} f (x) d x$

Step 3.2.2

Subtract $f (x) t$ from $0$ .

$lim_{t \to - \infty} - f (x) t + lim_{t \to \infty} \int_{0}^{t} f (x) d x$

Step 4

Apply the constant rule.

$lim_{t \to - \infty} - f (x) t + lim_{t \to \infty} f (x) x]_{0}^{t}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $f (x) x$ at $t$ and at $0$ .

$lim_{t \to - \infty} - f (x) t + lim_{t \to \infty} (f (x) t) - f (x) \cdot 0$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $0$ by $- 1$ .

$lim_{t \to - \infty} - f (x) t + lim_{t \to \infty} f (x) t + 0 f (x)$

Step 5.2.2

Multiply $0$ by $f (x)$ .

$lim_{t \to - \infty} - f (x) t + lim_{t \to \infty} f (x) t + 0$

Step 5.2.3

Add $f (x) t$ and $0$ .

$lim_{t \to - \infty} - f (x) t + lim_{t \to \infty} f (x) t$

Step 6

Evaluate the limits.

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Step 6.1

Reorder the factors of $- f (x) t$ .

$lim_{t \to - \infty} - t f (x) + lim_{t \to \infty} f (x) t$

Step 6.2

The limit at negative infinity of a polynomial of odd degree whose leading coefficient is negative is infinity.

$\infty + lim_{t \to \infty} f (x) t$

Step 6.3

Reorder the factors of $f (x) t$ .

$\infty + lim_{t \to \infty} t f (x)$

Step 6.4

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\infty + \infty$

Step 6.5

Infinity plus infinity is infinity.

$\infty$