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| Indefinite Integrals as Antiderivative |
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| Consider the following example: |
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Let f(x) = cos 3x, let us find a function F(x) such that  |
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We know that  |
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Here  |
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In other words we say the integral cos 3x is  |
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Suppose  then also we have |
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| Let us define integral of a function in general as follows. |
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| Let F(x) be a function such that |
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| \ In general, integral of f(x) is F(x) + C |
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| Where C is called the constant of integration. |
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| Let f(x) = 3x2 |
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| Note that for different values of C we get different integrals. But all these integrals are very similar geometrically. |
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| The function y = x3 + C represent a family of integrals. The above figure shows different curves of the integral function y = x3 + C. These curves fill the co-ordinate plane without overlapping. These curves together constitute the indefinite integrals. |
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If we draw a line x = a perpendicular to X-axis. Then the curves y = x3 + C have slopes  The slopes of the tangent at P1, P2, P3, P4 and P5 are equal (equal to 3a2). This indicates, the tangents to these curves are parallel at these points. |
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