    # Antiderivatives

In calculus, an antiderivative or indefinite integral, G, of a function g, is the function that can be differentiated to obtain the original function, that is, G’ = f.

G(x) = $\int g(x)dx$

# Examples

• $\int sinxdx = -cosx + C$ where C is a constant. $\int sec^{2}xdx = tanx + C. \int \frac{dx}{\sqrt{1-x^{2}}}=sin^{-1}x+C. \int \frac{dx}{1+x^{2}}=tan^{-1}x+C.$
• $\int \frac{1}{a+1}x^{a+1}dx + C$ where a≠-1 because d(xa+1)=(a+1)xadx.
• $\int \frac{dx}{x} = ln|x| + C.$ It is obvious for x positive. Let be x < 0, $\frac{d}{dx}ln|x|=\frac{d}{dx}ln(-x)=\frac{1}{-x}(-1)=\frac{1}{x}$
• $\int x^{3}(x^{4}+2)^{5}dx = \int u^{5}\frac{du}{4}$ where u = x4 + 2, du = 4x3dx.

$\int u^{5}\frac{du}{4} = \frac{u^{6}}{24} + C = \frac{(x^{4}+2)^{6}}{24} + C.$

• $\int \frac{dx}{xlnx} = \int u^{-1}du$ where u = lnx, du =$\frac{dx}{x}$

$\int u^{-1}du = ln|u| + C = ln|lnx| + C.$

• $\int \frac{x}{1+x^{2}}dx=\sqrt{1+x^{2}} + C.$

• $\int xe^{-x^{2}}dx = \frac{-1}{2}e^{-x^{2}} + C.$

$\frac{d}{dx}(1+x^{2})^{1/2} = \frac{1}{2}(1+x^{2})^{-1/2}(2x)$

• $\int e^{7x}dx = \frac{1}{7}e^{7x} + C.$
• $\int sinxcosxdx = \frac{1}{2}sin^{2}x + C_1 = \frac{-1}{2}cos^{2}x + C_2.$

# Uniqueness of Antiderivatives

Let F and G tow different antiderivatives of f(x). If F’ = G’, then F(x) = G(x) + C. They are unique up to a constant.

Proof. If F’ = G’ ⇒ (F-G)’ = F’ - G’ = 0 ⇒ F-G is constant ⇒ F(x) = G(x) + C.

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