Integrate 6(-1240/(x+1)+340) from 0 to 5

\int_{0}^{5}6\left(\left(-1\right)\cdot\frac{240}{x+1}+340\right)dx

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Answer

$7619.8664$

Step by step solution

Problem

$\int_{0}^{5}6\left(\left(-1\right)\cdot\frac{240}{x+1}+340\right)dx$
1

Taking the constant out of the integral

$6\int_{0}^{5}\left(340-\frac{240}{1+x}\right)dx$
2

The integral of a sum of two or more functions is equal to the sum of their integrals

$6\left(\int_{0}^{5}340dx+\int_{0}^{5}-\frac{240}{1+x}dx\right)$
3

The integral of a constant is equal to the constant times the integral's variable

$6\left(340x+\int_{0}^{5}-\frac{240}{1+x}dx\right)$
4

Taking the constant out of the integral

$6\left(340x-\int_{0}^{5}\frac{240}{1+x}dx\right)$
5

Apply the formula: $\int\frac{n}{b+x}dx$$=n\ln\left|b+x\right|$, where $b=1$ and $n=240$

$\left[6\left(340x-1\cdot 240\ln\left|1+x\right|\right)\right]_{0}^{5}$
6

Multiply $240$ times $-1$

$\left[6\left(340x-240\ln\left|1+x\right|\right)\right]_{0}^{5}$
7

Evaluate the definite integral

$\left(5\cdot 340+\ln\left|1+5\right|\left(-240\right)\right)\cdot 6-1\cdot \left(0\cdot 340+\ln\left|1+0\right|\left(-240\right)\right)\cdot 6$
8

Any expression multiplied by $0$ is equal to $0$

$\left(5\cdot 340+\ln\left|1+5\right|\left(-240\right)\right)\cdot 6-1\cdot \left(0+\ln\left|1+0\right|\left(-240\right)\right)\cdot 6$
9

Add the values $5$ and $1$

$\left(5\cdot 340+\ln\left|6\right|\left(-240\right)\right)\cdot 6-1\cdot \left(0+\ln\left|1\right|\left(-240\right)\right)\cdot 6$
10

Multiply $340$ times $5$

$\left(0+\ln\left|1\right|\left(-240\right)\right)\left(-6\right)+\left(1700+\ln\left|6\right|\left(-240\right)\right)\cdot 6$
11

Calculating the absolute value of $6$

$\left(0+\ln\left(1\right)\left(-240\right)\right)\left(-6\right)+\left(1700+\ln\left(6\right)\left(-240\right)\right)\cdot 6$
12

Calculating the natural logarithm of $6$

$\left(0+0\left(-240\right)\right)\left(-6\right)+\left(1700+1.7918\left(-240\right)\right)\cdot 6$
13

Any expression multiplied by $0$ is equal to $0$

$\left(0+0\right)\left(-6\right)+\left(1700+1.7918\left(-240\right)\right)\cdot 6$
14

Add the values $0$ and $0$

$0\left(-6\right)+\left(1700+1.7918\left(-240\right)\right)\cdot 6$
15

Any expression multiplied by $0$ is equal to $0$

$0+\left(1700+1.7918\left(-240\right)\right)\cdot 6$
16

Multiply $-240$ times $\sqrt{3}$

$0+\left(1700-430.0223\right)\cdot 6$
17

Subtract the values $1700$ and $-430.0223$

$0+1269.9777\cdot 6$
18

Multiply $6$ times $1269.9777$

$0+7619.8664$
19

Add the values $7619.8664$ and $0$

$7619.8664$

Answer

$7619.8664$

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.36 seconds

Views:

83