Question

The integral ∫13+5ex−6e−xdx. To solve this integral, you can use the substitution method. Let u=ex, and du=exdx. Then, the integral becomes:

∫13+5u−u61du
Now, you can find the antiderivative and add the constant of integration C to get the final result.

Answer 1

The final result of the integral ∫13+5ex−6e−xdx using the substitution method is ∫13+5u−u61du = 13u + 52 ln|u| - 16 ln|u| + C, where u = ex and C is the constant of integration.

Step-by-step explanation:

To solve the given integral using the substitution method, we first let u = ex, and then find du = exdx. Substituting these values into the original integral, we get ∫13+5u−u61du. Now, we can find the antiderivative of this expression. Integrating term by term, we get 13∫du + 5∫udu - ∫u6du. This simplifies to 13u + 52 ln|u| - 16 ln|u| + C, where C is the constant of integration.

The antiderivative of the given integral is obtained by substituting back u = ex into the expression. Therefore, the final result of the integral is 13ex + 52 ln|ex| - 16 ln|ex| + C. This can be further simplified to 13ex + 52x - 16x + C = 13ex + 36x + C.

By using the substitution method and finding the antiderivative, we have successfully solved the given integral to obtain the final result as 13ex + 36x + C.

Answer 2

The integral
`\(\int (3 + 5e^x - 6e^(-x)) \, dx\)` can be solved using the substitution method. Letting
`\(u = e^x\) and \(du = e^x \, dx\)`, the integral transforms into
`\(\int (3 + 5u - u^6) \, du\)`. By finding the antiderivative of this expression and adding the constant of integration (C), the final result is obtained.

Step-by-step explanation:

To solve the integral, the substitution method is employed by letting
`\(u = e^x\)`. The differential (du) is then calculated as \(e^x \, dx\). Substituting these values, the original integral
`\(\int (3 + 5e^x - 6e^(-x)) \, dx\)` becomes
`\(\int (3 + 5u - u^6) \, du\)`. This substitution simplifies the expression, making it more amenable to integration.

The antiderivative of the simplified expression is found by applying standard rules of integration. The result is
`\((1)/(7)u^7 + (5)/(2)u^2 - (u^7)/(7) + C\)`, where (C) is the constant of integration. Substituting back \(u = e^x\) yields the final antiderivative
`\((1)/(7)e^(7x) + (5)/(2)e^(2x) - (e^(7x))/(7) + C\).` This represents the solution to the original integral.

In conclusion, the substitution method simplifies the integral by introducing a new variable (u) and its differential (du). The antiderivative of the simplified expression is then found, and by substituting back the original variable, the solution to the integral is obtained, including the constant of integration (C).