Integration with U Substitution

Front

What is u-substitution in integration?

Back

U-substitution is a method used to simplify the process of integration by substituting a part of the integrand with a new variable, usually denoted as 'u', to make the integral easier to solve.

Front

When is an object moving to the left based on its velocity graph?

Back

An object is moving to the left when its velocity is negative. This can be determined by analyzing the intervals where the velocity function is less than zero.

Front

What is the derivative of lnx?

Back

The derivative of lnx is \( \frac{1}{x} \).

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Back

Front

What is the purpose of definite integrals?

Back

Definite integrals are used to calculate the net area under a curve between two specified limits, providing a numerical value that represents the accumulation of quantities.

Front

How do you determine the limits of integration in a definite integral?

Back

The limits of integration are determined by the specific interval over which the area or quantity is being calculated, often given in the problem statement.

Front

What is the integral of e^x?

Back

The integral of e^x is e^x + C.

Front

What does the constant 'C' represent in indefinite integrals?

Back

The constant 'C' represents the family of antiderivatives, accounting for all possible vertical shifts of the function.

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Back

Let u = 2x^3 + 1, then du = 6x^2 dx. Rewrite the integral in terms of u and solve.

Front

What is the first step in solving an integral using u-substitution?

Back

Identify a part of the integrand to substitute with 'u' and compute its derivative to replace dx.