Integrals

Integrals serve as key components in the field of calculus, typically utilized for determining areas, volumes, and various other values. The basic idea behind integrals is the accumulation of limitless infinitesimal quantities. There are two categories: definite integrals, responsible for computing the total area beneath a curve, and indefinite integrals, which function as the inverse of differentiation.

Finding the Integral

Determine the integral of the given function:

\int t^{3} {(t^{4} - 6)}^{4} d t =

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Evaluating Definite Integrals

Evaluate the following integral.

\int_{\ln (3 π / 2)}^{\ln (11 π / 6)} 10 e^{v} \cos e^{v} d v

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Evaluating Indefinite Integrals

Evaluate the following indefinite integral.

\int \frac{10}{\sqrt{x}} d x

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Substitution Rule

Evaluate the integral:

\int (3 x^{2} - 2 x + 1) e^{x^{3} - x^{2} + x} d x

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Rewriting as a Single Interval

Suppose

f (x) = {\begin{cases} x^{2} - 4 x for x < - 4 \\ 4 x - x^{2} for x \geq - 4 \end{cases}

then

\int_{- 16}^{4} f (x) d x

is equal to...

\int_{- 16}^{- 4} (x^{2} - 4 x) d x + \int_{- 4}^{4} (4 x - x^{2}) d x

\int_{- 16}^{0} (x^{2} - 4 x) d x + \int_{0}^{4} (4 x - x^{2}) d x

\int_{- 16}^{- 4} (4 x - x^{2}) d x + \int_{- 4}^{4} (x^{2} - 4 x) d x

\int_{- 16}^{- 4} (x^{2} - 4 x) d x + \int_{- 4}^{4} (x^{2} - 4 x) d x

\int_{- 16}^{0} (4 x - x^{2}) d x + \int_{0}^{4} (x^{2} - 4 x) d x

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Integrals

Finding the Integral

Evaluating Definite Integrals

Evaluating Indefinite Integrals

Substitution Rule

Rewriting as a Single Interval

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