Solve ∫ (6sinx+6x^6) wrt x

$$\int ( 6 \sin x + 6 x ^ { 6 } ) d x$$

Evaluate

$$-6\cos(x)+\frac{6x^{7}}{7}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 6\sin(x)\mathrm{d}x+\int 6x^{6}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$6\left(\int \sin(x)\mathrm{d}x+\int x^{6}\mathrm{d}x\right)$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply 6 times -\cos(x).

$$6\left(-\cos(x)+\int x^{6}\mathrm{d}x\right)$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{6}\mathrm{d}x with \frac{x^{7}}{7}. Multiply 6 times \frac{x^{7}}{7}.

$$-6\cos(x)+\frac{6x^{7}}{7}$$

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.

$$-6\cos(x)+\frac{6x^{7}}{7}+С$$

Differentiate w.r.t. x

$$6\left(\sin(x)+x^{6}\right)$$