Solve ∫ (e^x-8x^7+3cosx) wrt x=

$$\int ( e ^ { x } - 8 x ^ { 7 } + 3 \cos x ) d x =$$

Evaluate

$$e^{x}+3\sin(x)-x^{8}+С$$

Solution Steps

Integrate the sum term by term.

$$\int e^{x}\mathrm{d}x+\int -8x^{7}\mathrm{d}x+\int 3\cos(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int e^{x}\mathrm{d}x-8\int x^{7}\mathrm{d}x+3\int \cos(x)\mathrm{d}x$$

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

$$e^{x}-8\int x^{7}\mathrm{d}x+3\int \cos(x)\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{7}\mathrm{d}x with \frac{x^{8}}{8}. Multiply -8 times \frac{x^{8}}{8}.

$$e^{x}-x^{8}+3\int \cos(x)\mathrm{d}x$$

Use \int \cos(x)\mathrm{d}x=\sin(x) from the table of common integrals to obtain the result.

$$e^{x}-x^{8}+3\sin(x)$$

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.

$$e^{x}-x^{8}+3\sin(x)+С$$

Differentiate w.r.t. x

$$e^{x}+3\cos(x)-8x^{7}$$