# Integration by Parts

TIPS FOR SOLVING INTEGRATING FUNCTIONS

• Integrals which are in the form
\int {f(x) \cdot g(x)} \ dx
• Process :
\int u v dx = u \int v - \int (u' \int v) dx
First preference of u by ILATE
I = Inverse trignometric functions
L = logarithmic function
A = Algebric function
T = Trigonometric function
E = Exponential function
• Integration of functions type by leibnitz Rule
∫ uv dx = uv1 - u'v2 + u'' v3 − u''' v4 + ----

TRICKS ON INTEGRALS

• \int x^{n} \log x dx = \frac{x^{n+1}}{n+1} \left[\log x - \frac{1}{n+1}\right] + c
• \int e^{x} \left(f(x) + f'(x)\right) dx = e^{x} f(x) + c
• \int e^{-x} \left(f(x) - f'(x)\right) dx = -e^{-x} f(x) + c
• \int e^{ax} \left(f(x) + \frac{f'(x)}{a}\right) dx = \frac{e^{ax} f(x)}{a} + c
• \int e^{ax} \sin (bx + c) dx = \frac{e^{ax}}{a^{2} + b^{2}} \left(a \sin (bx + c) - b \cos (bx + c)\right) + k
• \int e^{ax} \cos (bx + c) dx = \frac{e^{ax}}{a^{2} + b^{2}} \left(a \cos (bx + c) + b \sin (bx + c) \right) + k
• \int a^{x} \sin (bx + c) dx = \frac{a^{x}}{(\log a)^{2} + b^{2}} \left[\log a \sin (bx + c) - b \cos (bx + c)\right] + k
• \int a^{x} \cos (bx + c) dx = \frac{a^{x}}{(\log a)^{2} + b^{2}} \left[\log a \cos (bx + c) + b \sin (bx + c)\right] + k
• \int xe^{ax} \sin (bx + c) dx = \frac{xe^{ax}}{r} \sin (bx + c - \theta) - \frac{e^{ax}}{r^{2}} \sin (bx + c - 2 \theta)
a = r cos θ, b = r sin θ, r = \sqrt{a^{2} + b^{2}}, \theta = \tan^{-1} \left(\frac{b}{a}\right)
• \int xe^{ax} \cos (bx + c) dx = \frac{xe^{ax}}{r} \cos (bx + c - \theta) \frac{-e^{ax}}{r^{2}} \cos (bx + c - 2 \theta)
where a = r cos b= r sin θ, \theta = \tan^{-1} \left(\frac{b}{a}\right)
• \int f^{'} (x) \left(f(x)\right)^{n} dx = \frac{f(x)^{n+1}}{n+1} + c
• \int \left(x f^{'} (x) + f (x) \right) dx = x f(x) + c
• ∫ f(x) dx = f(x) and g(x) is a differentialble function then ∫ (fog) (x) g'(x) dx = F (g(x) + c

### Part2: View the Topic in this video From 00:40 To 56:16

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• \int \sqrt{a^{2} + x^{2}} dx = \frac{x}{2} \sqrt{a^{2} + x^{2}} + \frac{a^{2}}{2} \log \begin{vmatrix} x + \sqrt{a^{2} + x^{2}}\end{vmatrix} + c
\int \sqrt{x^{2} - a^{2}} dx = \frac{x}{2} \sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2} \log \begin{vmatrix} x + \sqrt{x^{2} - a^{2}}\end{vmatrix} + c
\int \sqrt{a^{2} - x^{2}} dx = \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{-1} \left(\frac{x}{a}\right) + c
• I_{n} = \int \sin^{n}x \ dx = \frac{-\sin^{n-1}x \cos x}{n} + \frac{n-1}{n} I_{n-2}
• \tt I_{n} = \int \cos^{n} x \ dx = \frac{\cos^{n-1}x \sin x}{n} + \frac{n-1}{n} I_{n-2}
• I_{n} = \int \tan^{n} x dx = \frac{\tan^{n-1} x}{n-1} - I_{n-2}
• I_{n} = \int \cot^{n} x dx = \frac{-\cot^{n-1} x}{n-1} - I_{n-2}
• I_{n} = \int \sec^{n} x dx = \frac{\sec^{n-2} x \tan x}{n-1} + \frac{n-2}{n-1} I_{n-2}
• I_{n} = \int cosec^{n} x dx = \frac{-cosec^{n-2} x \cot x}{n-1} + \frac{n-2}{n-1} I_{n-2}
• I_{m,n} = \int {\sin^{m}_{x}} {\cos^{n}_{x}} d_{x}
• \begin{cases} \frac{\sin^{m+1} x \cos^{n-1} x}{m+n} \ + \ \frac{n-1}{m+n} \ I_{m, n-2} \\ \frac{-\sin^{m-1} x \cos^{n+1} x}{m+n} \ + \ \frac{m-1}{m+n} \ I_{m-2, n} \end{cases}
• I_{n} = \int x^{n} e^{ax} dx = \frac{e^{ax}x^{n}}{a} - \frac{n}{a} I_{n+1}
• I_{n} = \int (\log x)^{n} dx = x (\log x)^{n} - n I_{n-1}
• I_{m,n} = \int x^{m} (\log x)^{n} dx
\int \frac{x^{m+1} (\log x)^{n}}{m+1} - \frac{n}{m+1} I_{m, n-1}