What is the integral of 4/(sqrt(81-(x+9)^2)) ?

The integral of 4/(sqrt(81-(x+9)^2)) is 4arcsin(1/9 x+1)+C

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integral of 4/(sqrt(81-(x+9)^2))

\int \frac{4}{81 - ( x + 9 ) ^{2}} d x

4 arcsin (\frac{1}{9} x + 1) + C

Solution steps

\int \frac{4}{81 - ( x + 9 ) ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{1}{81 - ( x + 9 ) ^{2}} d x

Apply u-substitution

: \int \frac{1}{81 - u ^{2}} d u

= 4 \cdot \int \frac{1}{81 - u ^{2}} d u

Apply Trigonometric Substitution

: \int 1 d v

= 4 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 4 \cdot 1 \cdot v

Substitute back

= 4 \cdot 1 \cdot arcsin (\frac{1}{9} (x + 9))

Simplify

4 \cdot 1 \cdot arcsin (\frac{1}{9} (x + 9)) : 4 arcsin (\frac{1}{9} x + 1)

= 4 arcsin (\frac{1}{9} x + 1)

Add a constant to the solution

= 4 arcsin (\frac{1}{9} x + 1) + C

y^{^{'}} + 7 y = 3

\int \frac{x - 4}{x ( x - 1 ) ^{2}} d x

x \to 2 lim (\frac{x - 2}{x - 2})

\int (yz e^{x z}) d x

What is the integral of 4/(sqrt(81-(x+9)^2)) ?
The integral of 4/(sqrt(81-(x+9)^2)) is 4arcsin(1/9 x+1)+C