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Indefinite Integrals

Integrals - 1 part
Integrals - 2 part
Integrals - 3 part
Integrals - 4 part

Definition of an Indefinite Integral

If $\frac{dy}{dx}=f(x)$ then y is the function whose derivative is f(x) and is called anti - derivative of f(x) or the imdefinite integral of f(x), denoted by ∫f(x) dx. Similarly if y = ∫f(u) du, then $\frac{dy}{du}=f(u)$ . Since the derivative of a constant is zero, all indefinite integrals differ by an arbitrary constant.

The process of finding an integral is called integration.

General Rules of Integration

In the following u, v , w are functions of x; a, b, p, q, n any constants, restricted if indicated; e = 2.71828... is the natural base of logarithms; ln u denotes the natural logarithm of u where it is assumed that u > 0 [in general, to extend formulas to cases where u > 0 as well, replace ln u by ln |u|]; all angles are in radians; all constants of integration are omitted but implied.

∫ a dx = ax

∫ af(x) = a ∫ f(x) dx

∫ (u ± v ± w ± ....)dx = ∫ u dx ± ∫ v dx ± ∫ w dx ± ....

∫ u dv = u.v - ∫ v du [Integration by parts]
For generalized integration by parts, see 14.48.

$\int_{}^{}f(ax)dx=\frac{1}{a}.\int_{}^{}f(u)du$

$\int_{}^{}F[f(x)]dx=\int_{}^{}[F(u)\frac{dx}{du}]du=\int_{}^{}\frac{F(u)}{f'(x)}du$ where u = f(x).

$\int_{}^{}u^ndu=\frac{u^{n+1}}{n+1}$ ; n ≠ -1. [For n = -1, see 14.8]

$\int_{}^{}\frac{du}{u}=lnu=ln|u|$ ; if u > 0 or ln(-u) if u < 0.

∫ e^u du = e^u

$\int_{}^{}a^udu=\int_{}^{}e^{u.lna}du=\int_{}^{}\frac{e^{u.lna}}{lna}=\frac{a^u}{lna }$ ; a > 0, a ≠ 1.

∫ sin u du = - cos u

∫ cos u du = sin u

∫ tan u du = ln sec u = - ln cos u

∫ cot u du = ln sin u

∫ sec u du = ln (sec u + tan u) = ln tan [(2u + π)/4]

∫ csc u du = ln (csc u - cot u) = ln tan (u/2)

∫ sec² u du = tan u

∫ csc² u du = - cot u

∫ tan² u du = tan u - u

∫ cot² u du = - cot u - u

$\int_{}^{}sin^2udu=\frac{u}{2}-\frac{sin2u}{4}=\frac{u-sinu.cosu}{2}$

$\int_{}^{}cos^2udu=\frac{u}{2}+\frac{sin2u}{4}=\frac{u+sinu.cosu}{2}$

∫ sec u.tan u du = sec u

∫ csc u.cot u du = - csc u

∫ sinh u du = cosh u

∫ cosh u du = sinh u

∫ tanh u du = ln cosh u

∫ coth u du = ln sinh u

∫ sech u du = sin^-1(tanh u) or 2tan^-1.e^u

∫ csch u du = ln tanh(u/2) or - coth^-1.e^u

∫ sech² u du = tanh u

∫ csch² u du = - coth u

∫ tanh² u du = u - tanh u

∫ coth² u du = u - coth u

$\int_{}^{}sinh^2udu=\frac{sinh2u}{4}-\frac{u}{2}=\frac{sinhu.coshu-u}{2}$

$\int_{}^{}cosh^2udu=\frac{sinh2u}{4}+\frac{u}{2}=\frac{sinhu.coshu+u}{2}$

∫ sech u.tanh u du = - sech u

∫ csch u.coth u du = - csch u

This is called generalized integration by parts.

Important Transformations

Often in practice an integral can be simplified by using an appropriate transformation or substitution and formula 14.6. The following list gives some transformations and their effects.

$\int_{}^{}F(ax+b)dx=\frac{1}{a}.\int_{}^{}F(u)du$ where u = ax + b

$\int_{}^{}F(\sqrt{ax+b})dx=\frac{2}{a}.\int_{}^{}u.F(u)du$ where u = √ax + b

∫ F(√a² - x²) dx = a∫ F(a.cos u).cos u du where x = a.sin u

∫ F(√x² + a²) dx = a∫ F(a.sec u).sec² u du where x = a.tan u

∫ F(√x² - a²) dx = a∫ F(a.tan u).sec u.tan u du where x = a.sec u

$\int_{}^{}F(e^{ax})dx=\frac{1}{a}.\int_{}^{}\frac{F(u)}{u}du$ where u = e^ax

∫ F(ln x) dx = ∫ F(u).e^u du where u = ln x

∫ F(sin^-1(x/a)) dx = a∫ F(u).cos u du where u = sin^-1(x/a)
Similar results apply for other inverse trigonometric functions.

∫ F(sin x, cos x) dx = 2

Special Integrals

Pages 60 through 93 provide a table of integrals classified under special types. The remarks given on page 57 apply here as well. It is assumed in all cases that division by zero is excluded.