Solve ∫ (e^x+4/x+5x^9+6/x^4*7x*8) wrt x

$$\int ( e ^ { x } + \frac { 4 } { x } + 5 x ^ { 9 } + \frac { 6 } { x ^ { 4 } } \cdot 7 x \cdot 8 ) d x$$

Evaluate

$$e^{x}+4\ln(|x|)+\frac{x^{10}}{2}-\frac{168}{x^{2}}+С$$

Solution Steps

Multiply 7 and 8 to get 56.

$$\int e^{x}+\frac{4}{x}+5x^{9}+\frac{6}{x^{4}}\times 56x\mathrm{d}x$$

Express \frac{6}{x^{4}}\times 56 as a single fraction.

$$\int e^{x}+\frac{4}{x}+5x^{9}+\frac{6\times 56}{x^{4}}x\mathrm{d}x$$

Express \frac{6\times 56}{x^{4}}x as a single fraction.

$$\int e^{x}+\frac{4}{x}+5x^{9}+\frac{6\times 56x}{x^{4}}\mathrm{d}x$$

Cancel out x in both numerator and denominator.

$$\int e^{x}+\frac{4}{x}+5x^{9}+\frac{6\times 56}{x^{3}}\mathrm{d}x$$

Multiply 6 and 56 to get 336.

$$\int e^{x}+\frac{4}{x}+5x^{9}+\frac{336}{x^{3}}\mathrm{d}x$$

Integrate the sum term by term.

$$\int e^{x}\mathrm{d}x+\int \frac{4}{x}\mathrm{d}x+\int 5x^{9}\mathrm{d}x+\int \frac{336}{x^{3}}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int e^{x}\mathrm{d}x+4\int \frac{1}{x}\mathrm{d}x+5\int x^{9}\mathrm{d}x+336\int \frac{1}{x^{3}}\mathrm{d}x$$

Use \int e^{x}\mathrm{d}x=e^{x} from the table of common integrals to obtain the result.

$$e^{x}+4\int \frac{1}{x}\mathrm{d}x+5\int x^{9}\mathrm{d}x+336\int \frac{1}{x^{3}}\mathrm{d}x$$

Use \int \frac{1}{x}\mathrm{d}x=\ln(|x|) from the table of common integrals to obtain the result.

$$e^{x}+4\ln(|x|)+5\int x^{9}\mathrm{d}x+336\int \frac{1}{x^{3}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{9}\mathrm{d}x with \frac{x^{10}}{10}. Multiply 5 times \frac{x^{10}}{10}.

$$e^{x}+4\ln(|x|)+\frac{x^{10}}{2}+336\int \frac{1}{x^{3}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{x^{3}}\mathrm{d}x with -\frac{1}{2x^{2}}. Multiply 336 times -\frac{1}{2x^{2}}.

$$e^{x}+4\ln(|x|)+\frac{x^{10}}{2}-\frac{168}{x^{2}}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$e^{x}+4\ln(|x|)+\frac{x^{10}}{2}-\frac{168}{x^{2}}+С$$

Differentiate w.r.t. x

$$e^{x}+5x^{9}+\frac{4}{x}+\frac{336}{x^{3}}$$