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Revision:Further Integration

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Further Integration

1 Hyperbolic Standard Integrals
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Hyperbolic Standard Integrals

$\displaystyle \int sinhx \hspace5 dx = coshx + C$

$\displaystyle \int coshx \hspace5 dx = sinhx + C$

$\displaystyle \int sech^2x \hspace5 dx = tanhx + C$

$\displaystyle \int cosech^2x = -cothx + C$

$\displaystyle \int cosechxcothx = -cosechx + C$

$\displaystyle \int sechx = ln sechx + tanhx + C$

$\displaystyle \int cosechx = ln tanh \frac{x}{2} + C$

$\displaystyle \int tanhx \hspace5 dx = ln sechx + C$

$\displaystyle \int cothx \hspace5 dx = ln sinhx + C$

Inverse Trig and Hyperbolic Integrals

$\displaystyle \int \frac{1}{\sqrt{x^2+a^2}} = arsinh\frac{x}{a} + C$

$\displaystyle \int \frac{1}{\sqrt{x^2-a^2}} = arcosh\frac{x}{a} + C$

$\displaystyle \int \frac{1}{1-x^2} = artanhx + C$

$\displaystyle \int \frac{1}{\sqrt{1-x^2}} = arcsinx + C$

$\displaystyle \int \frac{-1}{\sqrt{1-x^2}} = arccosx + C$

$\displaystyle \int \frac{1}{a^2 +x^2} = \frac{1}{a}arctan\frac{x}{a} + C$

$\displaystyle \int \frac{1}{x^2-a^2} = \frac{1}{2a} ln |\frac{x-a}{x+a}| + C$

$\displaystyle \int \frac{1}{a^2-x^2}= \frac{1}{2a} ln |\frac{a+x}{a-x}| + C$

Reduction Formula

The Reduction Formula is used when integrals can be reduced by generalising the integral for a function to the powers of n. From this you can substitute a value for n and evaluate a definite or indefinite integral relatively easy. They are especially useful when integrating large powers of trigonometric functions.

For example :

Find $\displaystyle \int x^3e^x \hspace5 dx$

take $I_n = \displaystyle \int x^ne^x \hspace5 dx$

The I_n refers to the integral where there is a power of n involved.

Evaluating this by integration by parts:

u = x^n

$\frac{du}{dx} = nx^{n-1}$

$\frac{dv}{dx} = e^x$

v = e^x

$\displaystyle I_n = x^ne^x - \int nx^{n-1}e^x$

$\displaystyle I_n = x^ne^x - n \int x^{n-1}e^x$

Notice that $\displaystyle \int x^{n-1}e^x = I_{n-1}$

becomes $I_n = x^ne^x - nI_{n-1}$

From this, we can find $\displaystyle \int x^3e^x \hspace5 dx$

by saying "let n = 3".

Therefore,

I_3 = x^3e^x - 3I_2

I_2 = x^2e^x - 2I_1

$I_1 = \displaystyle \int xe^x$

u = x

$\frac{du}{dx} = 1$

$\frac{dv}{dx} = e^x$

v = e^x

$I_1 = xe^x - \displaystyle \int e^x$

I_1 = xe^x-e^x + C

Therefore,

I_3 = x^3e^x -3x^2e^x +6(xe^x-e^x) + C

I_3 = x^3e^x - 3x^2e^x +6xe^x- 6e^x + C

Therefore,

$\displaystyle \int x^3e^x \hspace5 dx = x^3e^x - 3x^2e^x + 9xe^x-9e^x + C$

This can be used for sin^nx and other trigonometric functions.

Length of a Curve

Arc Length AB = $\displaystyle \int_{x_a}^{x_b} \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{1}{2}} \hspace5 dx$

Arc Length AB = $\displaystyle \int_{x_a}^{x_b} \left[1+\left(\frac{dx}{dy}\right)^2\right]^{\frac{1}{2}} \hspace5 dx$

Arc Length AB = $\displaystyle \int_{x_a}^{x_b} \left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right]^{\frac{1}{2}} \hspace5 dt$

Area of Surface of Revolution

When rotated around the x axis:

Surface Area = $\displaystyle \int_{x_a}^{x_b} \displaystyle 2\pi y \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{1}{2}} \hspace5 dx$

When using parametric coordinates (for rotation around the x axis):

Surface Area = $\displaystyle \int_{x_a}^{x_b} 2\pi y \left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right]^{\frac{1}{2}} \hspace5 dt$