What is the integral of 1/((x+7)(49x^2+7x+1)) ?

The integral of 1/((x+7)(49x^2+7x+1)) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/((x+7)(49x^2+7x+1))

\int \frac{1}{( x + 7 ) ( 49 x ^{2} + 7 x + 1 )} d x

56 (\frac{1}{131768} ln ∣ 14 x + 98 ∣ + \frac{1}{9412} (- \frac{1}{28} ln 196 x^{2} + 28 x + 4 + \frac{97}{14 3} arctan (\frac{14 x + 1}{3}))) + C

Solution steps

\int \frac{1}{( x + 7 ) ( 49 x ^{2} + 7 x + 1 )} d x

Apply u-substitution

: \int \frac{1}{u ( 49 u ^{2} - 679 u + 2353 )} d u

= \int \frac{1}{u ( 49 u ^{2} - 679 u + 2353 )} d u

Complete the square

49 u^{2} - 679 u + 2353 : 49 (u - \frac{97}{14})^{2} + \frac{3}{4}

= \int \frac{1}{u ( 49 ( u - \frac{97}{14} ) ^{2} + \frac{3}{4} )} d u

Apply u-substitution

: \int \frac{56}{( 14 v + 97 ) ( 196 v ^{2} + 3 )} d v

= \int \frac{56}{( 14 v + 97 ) ( 196 v ^{2} + 3 )} d v

Take the constant out:

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

= 56 \cdot \int \frac{1}{( 14 v + 97 ) ( 196 v ^{2} + 3 )} d v

Take the partial fraction of

\frac{1}{( 14 v + 97 ) ( 196 v ^{2} + 3 )} : \frac{1}{9412 ( 14 v + 97 )} + \frac{- 14 v + 97}{9412 ( 196 v ^{2} + 3 )}

= 56 \cdot \int \frac{1}{9412 ( 14 v + 97 )} + \frac{- 14 v + 97}{9412 ( 196 v ^{2} + 3 )} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 56 (\int \frac{1}{9412 ( 14 v + 97 )} d v + \int \frac{- 14 v + 97}{9412 ( 196 v ^{2} + 3 )} d v)

\int \frac{1}{9412 ( 14 v + 97 )} d v = \frac{1}{131768} ln ∣ 14 v + 97 ∣

\int \frac{- 14 v + 97}{9412 ( 196 v ^{2} + 3 )} d v = \frac{1}{9412} (- \frac{1}{28} ln 196 v^{2} + 3 + \frac{97}{14 3} arctan (\frac{14}{3} v))

= 56 (\frac{1}{131768} ln ∣ 14 v + 97 ∣ + \frac{1}{9412} (- \frac{1}{28} ln 196 v^{2} + 3 + \frac{97}{14 3} arctan (\frac{14}{3} v)))

Substitute back

= 56 (\frac{1}{131768} ln 14 (x + 7 - \frac{97}{14}) + 97 + \frac{1}{9412} (- \frac{1}{28} ln 196 (x + 7 - \frac{97}{14})^{2} + 3 + \frac{97}{14 3} arctan (\frac{14}{3} (x + 7 - \frac{97}{14}))))

Simplify

56 (\frac{1}{131768} ln 14 (x + 7 - \frac{97}{14}) + 97 + \frac{1}{9412} (- \frac{1}{28} ln 196 (x + 7 - \frac{97}{14})^{2} + 3 + \frac{97}{14 3} arctan (\frac{14}{3} (x + 7 - \frac{97}{14})))) : 56 (\frac{1}{131768} ln ∣ 14 x + 98 ∣ + \frac{1}{9412} (- \frac{1}{28} ln 196 x^{2} + 28 x + 4 + \frac{97}{14 3} arctan (\frac{14 x + 1}{3})))

= 56 (\frac{1}{131768} ln ∣ 14 x + 98 ∣ + \frac{1}{9412} (- \frac{1}{28} ln 196 x^{2} + 28 x + 4 + \frac{97}{14 3} arctan (\frac{14 x + 1}{3})))

Add a constant to the solution

= 56 (\frac{1}{131768} ln ∣ 14 x + 98 ∣ + \frac{1}{9412} (- \frac{1}{28} ln 196 x^{2} + 28 x + 4 + \frac{97}{14 3} arctan (\frac{14 x + 1}{3}))) + C

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

\int \frac{( x + 3 )}{x ^{2} + 6 x} d x

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

\int 2^{1 - 2 x} d x

\int sec^{4} (θ) tan^{2} (θ) d θ

What is the integral of 1/((x+7)(49x^2+7x+1)) ?
The integral of 1/((x+7)(49x^2+7x+1)) is