Solve ∫ (-2/3ab^4+3/4ab+5a^3)+(-1/2ab+5/2a^3)-(-frac{11}{4}ab+frac{2}{2}ab^4)

$$\int ( - \frac { 2 } { 3 } a b ^ { 4 } + \frac { 3 } { 4 } a b + 5 a ^ { 3 } ) + ( - \frac { 1 } { 2 } a b + \frac { 5 } { 2 } a ^ { 3 } ) - ( - \frac { 11 } { 4 } a b + \frac { 2 } { 2 } a b ^ { 4 } )$$

Evaluate

$$\frac{ax\left(18b+45a^{2}-10b^{4}\right)}{6}+С$$

Solution Steps

Divide 2 by 2 to get 1.

$$\int -\frac{2}{3}ab^{4}+\frac{3}{4}ab+5a^{3}-\frac{1}{2}ab+\frac{5}{2}a^{3}-\left(-\frac{11}{4}ab+1ab^{4}\right)\mathrm{d}x$$

Combine \frac{3}{4}ab and -\frac{1}{2}ab to get \frac{1}{4}ab.

$$\int -\frac{2}{3}ab^{4}+\frac{1}{4}ab+5a^{3}+\frac{5}{2}a^{3}-\left(-\frac{11}{4}ab+1ab^{4}\right)\mathrm{d}x$$

Combine 5a^{3} and \frac{5}{2}a^{3} to get \frac{15}{2}a^{3}.

$$\int -\frac{2}{3}ab^{4}+\frac{1}{4}ab+\frac{15}{2}a^{3}-\left(-\frac{11}{4}ab+1ab^{4}\right)\mathrm{d}x$$

To find the opposite of -\frac{11}{4}ab+1ab^{4}, find the opposite of each term.

$$\int -\frac{2}{3}ab^{4}+\frac{1}{4}ab+\frac{15}{2}a^{3}+\frac{11}{4}ab-ab^{4}\mathrm{d}x$$

Combine \frac{1}{4}ab and \frac{11}{4}ab to get 3ab.

$$\int -\frac{2}{3}ab^{4}+3ab+\frac{15}{2}a^{3}-ab^{4}\mathrm{d}x$$

Combine -\frac{2}{3}ab^{4} and -ab^{4} to get -\frac{5}{3}ab^{4}.

$$\int -\frac{5}{3}ab^{4}+3ab+\frac{15}{2}a^{3}\mathrm{d}x$$

Find the integral of -\frac{5ab^{4}}{3}+3ab+\frac{15a^{3}}{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(-\frac{5ab^{4}}{3}+3ab+\frac{15a^{3}}{2}\right)x$$

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.

$$\left(\frac{15a^{3}}{2}-\frac{5ab^{4}}{3}+3ab\right)x+С$$

Differentiate w.r.t. x

$$-\frac{5ab^{4}}{3}+\frac{15a^{3}}{2}+3ab$$