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represents the differential coefficient of 
, we need to find 
or 
.
used to show the integration, it is a symbol of 
derived from the word 
. i.e. Integration is a process in which we have to sum up the derivatives over a specified interval and to find the function.
, when 
or 
is our solution![\[ \int dy = \int x^n dx \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-5eda050b0430a01906e860efef8149a4_l3.png "Rendered by QuickLaTeX.com")
![\[ y = \frac{x^{n+1}}{n+1} + C \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-8a2a9aff2b96da4579336e6de3f2b2de_l3.png "Rendered by QuickLaTeX.com")
:
“.
, 
, 
, 
the derivatives of all of them is 
, in finding the anti derivative of 
”, as we don’t know which constant was present in the original function, and can be found If we have initial boundary values (Definite Integral).
![\[ \int dy = \int x^3 dx \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-b54d41a7c8a86e438ce9d9ce1d8c5c6c_l3.png "Rendered by QuickLaTeX.com")
![\[ y = \frac{x^{3+1}}{3+1} + C \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-88c56df74fc1721f5823fa7c260a2a31_l3.png "Rendered by QuickLaTeX.com")
![\[ y = \frac{x^{4}}{4} + C \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-b2d1a07e624472c515d73d29a224645c_l3.png "Rendered by QuickLaTeX.com")
![\[ \int (x^3 + x^2 + 5x + 6) \, dx \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-9bfebde6ee2480834165f056e433841f_l3.png "Rendered by QuickLaTeX.com")
![\[ \int x^3 \, dx + \int x^2 \, dx + \int 5x \, dx + \int 6 \, dx \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-4a364089651e22547ef20000162fc837_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{x^4}{4} + \frac{x^3}{3} + \frac{5x^2}{2} + 6x + C \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-c3db210d63c8b1230e6fcbd1a1493f6f_l3.png "Rendered by QuickLaTeX.com")
![\[ z = a + bi \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-4b9d086002f2d7b1c48c0f576c4a38bf_l3.png "Rendered by QuickLaTeX.com")
is the real part of the complex number,
is the imaginary part,
is the imaginary unit, defined as 
.
, 
is the real part and 
is the imaginary part.![\[ (a + bi) + (c + di) = (a + c) + (b + d)i \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-a364308332b896c378c4acd09427f6f0_l3.png "Rendered by QuickLaTeX.com")
![\[ (a + bi) - (c + di) = (a - c) + (b - d)i \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-eae4f32ce38de845b2074491d90d9b3e_l3.png "Rendered by QuickLaTeX.com")
.![\[ (a + bi)(c + di) = (ac - bd) + (ad + bc)i \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-06987f4d02cfe7d931611ee1058e6d75_l3.png "Rendered by QuickLaTeX.com")
![\[ \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-7c5d34079dd5b2d0d69861d531278211_l3.png "Rendered by QuickLaTeX.com")
is 
. Conjugates are useful in division and finding magnitudes.![\[ Conjugate of (a + bi) = a - bi \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-5d2ffd06e13f1c1cce4727c767c536be_l3.png "Rendered by QuickLaTeX.com")
![\[ |a + bi| = \sqrt{a^2 + b^2} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-715f6163933a79f1ae398b17964580a8_l3.png "Rendered by QuickLaTeX.com")
such that:![\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-a653df83a37f4a3e21e7bb7e3a1729b0_l3.png "Rendered by QuickLaTeX.com")
![\[z = a + bi\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-3d72a2203045b4d2c2eddd8a26267657_l3.png "Rendered by QuickLaTeX.com")
can be expressed in polar form as:![\[ z = r \left( \cos \theta + i \sin \theta \right) \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-e3ddabc187698e2459026023848aa02e_l3.png "Rendered by QuickLaTeX.com")
is the magnitude (or modulus) of the complex number, given by![\[ r = |z| = \sqrt{x^2 + y^2} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-f95e0c37f2d0bf07dab2f34e7cb53726_l3.png "Rendered by QuickLaTeX.com")
-axis, given by![\[ \theta = \arg(z) = \tan^{-1} \left( \frac{y}{x} \right) \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-1183d088db87a9cde04ee36f27c3d410_l3.png "Rendered by QuickLaTeX.com")
![\[ z = r e^{i \theta} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-37a71f5876e61a5d27fb53e5c7eaeb5e_l3.png "Rendered by QuickLaTeX.com")
![\[ e^{i \theta} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-f37e5c80ddad9593bb9fbc9eed51f34f_l3.png "Rendered by QuickLaTeX.com")
![\[ z = 3 + 4i \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-a8ce511b6593b428905c27c385f38b71_l3.png "Rendered by QuickLaTeX.com")
![\[ r = \sqrt{3^2 + 4^2} = 5 \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-d686032da2f621b1d4dbe542f9dcbf07_l3.png "Rendered by QuickLaTeX.com")
![\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-61c7ebd90ef5428825045dd2509dfa21_l3.png "Rendered by QuickLaTeX.com")
![\[ z = 5 \left(\cos{53.13^\circ} + i \sin{53.13^\circ}\right) \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-b787afef3c263a92f115b4426a5077a2_l3.png "Rendered by QuickLaTeX.com")
![\[ z = 5 e^{i 53.13^\circ} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-9b20d26a5849831ac9a20846f91f0717_l3.png "Rendered by QuickLaTeX.com")
![\[ z = -1 + i \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-9d1c5e5eaaf0f316a0297a75eaa1c69f_l3.png "Rendered by QuickLaTeX.com")
![\[ r = \sqrt{-1^2 + 1^2} = \sqrt{2} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-5b953f44894e3bf3e47a3b1bf7e79318_l3.png "Rendered by QuickLaTeX.com")
![\[\theta = \tan^{-1}(1 / -1) = 135^\circ \, \text{or} \, \pi=\ - \frac{\pi}{4}\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-5f997691360254f9ce21f268a9e1e569_l3.png "Rendered by QuickLaTeX.com")
![\[z = \sqrt{2} (\cos{135^\circ} + i \sin{135^\circ})\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-85f2b6dc13a8853118b4fd39ea6f6c73_l3.png "Rendered by QuickLaTeX.com")
![\[ z = \sqrt{2} e^{i 135^\circ} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-5e10ecb4c26f5b3cc432b12300e90f33_l3.png "Rendered by QuickLaTeX.com")
![\[z=r(cosθ+isinθ)\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-54de396de9d0b3ca5c5323750fe98d4f_l3.png "Rendered by QuickLaTeX.com")
![\[z = cosθ+isinθ \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-7f7bfe1f57132b0c6a9d30007418073f_l3.png "Rendered by QuickLaTeX.com")
![\[z = e^{i\theta}\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-5dc90684e5c620ec6bfa9f6fae044364_l3.png "Rendered by QuickLaTeX.com")
![\[z = r e^{i\theta}\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-3ce81f6a2495c6246d8a19c8323ecde9_l3.png "Rendered by QuickLaTeX.com")
![\[e^{i\theta}\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-434e1df7a8627e2dc646cd83b5856ea6_l3.png "Rendered by QuickLaTeX.com")
![\[cos{\theta} + i \sin{\theta}\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-437ffcda5482e8fcb94032f0af32d069_l3.png "Rendered by QuickLaTeX.com")
![\[e^{i\theta} = \sum_{n=0}^{\infty} \frac{(i\theta)^n}{n!}\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-26fdba68f93d661f564ef526eff30a2b_l3.png "Rendered by QuickLaTeX.com")
![\[(i\theta)^n\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-0375746f2920913fd082d50fa081a2aa_l3.png "Rendered by QuickLaTeX.com")
![\[i^n\theta^n\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-4392905acb22346fdb7760a9a417376b_l3.png "Rendered by QuickLaTeX.com")
![\[e^{i\theta} = \sum_{n=0}^{\infty} \frac{i^n\theta^n}{n!}\]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-429a7b0dd717f96005c053a2c22e9eae_l3.png "Rendered by QuickLaTeX.com")
![\[ i^n = \begin{cases} 1 \quad \text{if } n \mod 4 = 0 \\ i \quad \text{if } n \mod 4 = 1 \\ -1 \quad \text{if } n \mod 4 = 2 \\ -i \quad \text{if } n \mod 4 = 3 \end{cases} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-86b22a40afad08e7534c3cda493a9a72_l3.png "Rendered by QuickLaTeX.com")
![\[ \text{Re}(e^{i\theta}) = \sum_{\text{n even}} \frac{(-1)^{n/2} \theta^n}{n!} = \cos{\theta} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-171d5c769159f9a3df2ad1e902bd2eef_l3.png "Rendered by QuickLaTeX.com")
![\[ \text{Im}(e^{i\theta}) = \sum_{\text{n odd}} \frac{(-1)^{(n-1)/2} \theta^n}{n!} = \sin{\theta} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-e36baffbdc9ce1f10bf6972fadc318e1_l3.png "Rendered by QuickLaTeX.com")
![\[ e^{i\theta} = \cos{\theta} + i \sin{\theta} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-3cd1ecd779f9b151ba86ae719b740175_l3.png "Rendered by QuickLaTeX.com")
into polar form.
into Euler’s form.
.
to Euler’s form.
in polar form.
.
into polar form.
to Euler’s form.
in polar form.
.
to Euler’s form.
in polar form.
.
into Euler’s form.
in polar form.
to Euler’s form.
.
to Euler’s form.
in polar form.
.
to polar form.
into Euler’s form.
.
to Euler’s form.
in polar form.
.
to Euler’s form.
in polar form.
.
to polar form.
, convert to Cartesian form.
into Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
to Cartesian form.
where 
are constants, and 
. The term “quadratic” comes from “quad,” meaning square, since the highest exponent of the variable is 
.
, ![\[ ax^2 + bx + c = (px + q)(rx + s) \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-362166951527e115ffb9c05ca0be9a21_l3.png "Rendered by QuickLaTeX.com")
and 
are constants, and the equation is solved by finding appropriate factors of 
.![\[ ax^2 + bx + c = a\left( x^2 + \frac{b}{a}x \right) + c \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-bfe624b353b3ae033612013c936cc777_l3.png "Rendered by QuickLaTeX.com")
![\[ x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-8759ed45590084aaf7bc61420b0f79aa_l3.png "Rendered by QuickLaTeX.com")
![\[ \left( x + \frac{b}{2a} \right)^2 - \text{constant} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-a3101fd9f5876b874d97c09a0daba6d2_l3.png "Rendered by QuickLaTeX.com")
![\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-406607c8b9fb91e3b47cc51fb4eb1348_l3.png "Rendered by QuickLaTeX.com")
, we check the sign of ![\[ x^2 + 7x + 12 = 0 \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-6220c4495e635f16b66525880512ad76_l3.png "Rendered by QuickLaTeX.com")
:![\[ (1, 12), (2, 6), (3, 4) \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-64c234a37ac61a78f99c827bcbc1634f_l3.png "Rendered by QuickLaTeX.com")
, we split 
into 
and 
, so the equation becomes:![\[ x^2 + 3x + 4x + 12 = 0 \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-0420fa30210e15fa84858a7f07b848a2_l3.png "Rendered by QuickLaTeX.com")
![\[ x(x + 3) + 4(x + 3) = 0 \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-6f4dc57d37b91937abb8a0865d2145eb_l3.png "Rendered by QuickLaTeX.com")
![\[ (x + 3)(x + 4) = 0 \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-6519df73e1914050125d4d6e549db1d3_l3.png "Rendered by QuickLaTeX.com")
![\[ x + 3 = 0 \quad \text{or} \quad x + 4 = 0 \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-41f1edaf99cd5767abd0ce1604b43c45_l3.png "Rendered by QuickLaTeX.com")
![\[ x = -3 \quad \text{and} \quad x = -4 \]](https://pakistanonlinetuition.com/wp-content/ql-cache/quicklatex.com-7b01a2877530ff34e12eda86258e947b_l3.png "Rendered by QuickLaTeX.com")
, in this case we need to find the factors of 
, so again the difference of 
, but we put 
sign with 
as 
. now the equation becomes
and 
and 
.
, repeat the case I and case II accordingly.
.
.
.
.
.
?
.
.
.
by factoring.
using the quadratic formula.
.
by completing the square.
.
using the quadratic formula.
.
.
by factoring.
?
using the quadratic formula.
.
:
into equation 
