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Power Rule for Antiderivatives

∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C, where n≠−1n \neq -1

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Power Rule for Antiderivatives
$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$
Antiderivative of $\sin(x)$
$\int \sin(x) dx = -\cos(x) + C$
Antiderivative of $\cos(x)$
$\int \cos(x) dx = \sin(x) + C$
Antiderivative of $\tan(x)$
$\int \tan(x) dx = -\ln|\cos(x)| + C$
Antiderivative of $\cot(x)$
$\int \cot(x) dx = \ln|\sin(x)| + C$
Antiderivative of $\sec(x)$
$\int \sec(x) dx = \ln|\sec(x) + \tan(x)| + C$
Antiderivative of $\csc(x)$
$\int \csc(x) dx = -\ln|\csc(x) + \cot(x)| + C$
Antiderivative of $e^x$
$\int e^x dx = e^x + C$
Derivative of $\arcsin(x)$
$\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$
Derivative of $\arctan(x)$
$\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2}$
Explain U-substitution.
Substitute part of the integrand with 'u', find du, rewrite the integral in terms of 'u', integrate, and substitute back to 'x'.
Explain the importance of '+ C'.
Represents the constant of integration because the derivative of a constant is zero.
When is long division useful in integration?
When integrating rational functions where the degree of the numerator is greater than or equal to the denominator.
Explain completing the square.
A method to rewrite a quadratic expression into a perfect square trinomial plus a constant, facilitating integration.
How do trigonometric identities help with integration?
They simplify complex trigonometric expressions into forms that are easier to integrate.
When to use U-substitution?
When the integrand contains a composite function and its derivative (or a multiple thereof).
What is the first step in U-substitution?
Choose a suitable 'u' that simplifies the integral.
What is the goal of selecting techniques for antidifferentiation?
To choose the correct method to find the antiderivative of a given expression.
What is the antiderivative of $\frac{1}{\sqrt{1-x^2}}$?
$\arcsin(x) + C$
What is the antiderivative of $\frac{1}{1+x^2}$?
$\arctan(x) + C$
Define antiderivative.
A function whose derivative is the given function.
What is the constant of integration?
The arbitrary constant 'C' added to the antiderivative.
Define U-substitution.
A technique for simplifying integrals by substituting a part of the integrand with a new variable 'u'.
What is a perfect square trinomial?
A trinomial that can be factored into the square of a binomial.
Define indefinite integral.
The family of all antiderivatives of a function.
What is the power rule for antiderivatives?
A rule to find the antiderivative of a power function.
What is long division used for in integration?
Simplifying rational functions where the degree of the numerator is greater than or equal to the degree of the denominator.
What is 'completing the square'?
A technique used to rewrite a quadratic expression into a perfect square trinomial plus a constant.
What is the purpose of trigonometric identities in integration?
To simplify trigonometric integrals into forms that are easier to evaluate.
What is the antiderivative of $\frac{1}{x}$?
$\ln|x| + C$