# Mathematics Questions and Answers – Integration as an Inverse Process of Differentiation

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This set of Mathematics Quiz for Engineering Entrance Exams focuses on “Integration as an Inverse Process of Differentiation”.

1. Find the integral of $$8x^3+1$$.
a) 2x4+x+C
b) 2x6-5x+C
c) 2x4-x+C
d) 2x4+x2 C

Explanation: $$\int \,8x^{3+1} \,dx$$
Using $$\int \,x^n \,dx=\frac{x^{n+1}}{n+1}$$, we get
$$\int \,8x^{3+1} \,dx=\int 8x^3 \,dx+\int \,1 \,dx$$
=$$\frac{8x^{3+1}}{3+1}+x$$
=$$\frac{8x^4}{4}+x$$
=2x4+x+C.

2. Find ∫ 7x2-x3+2x dx.
a) $$\frac{7x^3}{3}+\frac{x^4}{5}-\frac{2x^2}{2}+C$$
b) $$\frac{7x^3}{3}+\frac{x^4}{4}+\frac{2x^2}{2}+C$$
c) $$\frac{7x^5}{9}-\frac{x^4}{4}+\frac{2x^2}{2}+C$$
d) $$\frac{7x^3}{3}-\frac{x^4}{4}+x^2+C$$

Explanation: To find $$\int 7x^2-x^3+2x dx$$
$$\int 7x^2-x^3+2x dx=\int 7x^2 dx-\int x^3 dx+2\int x dx$$
Using $$\int x^n dx=\frac{x^{n+1}}{n+1}$$, we get
$$\int 7x^2-x^3+2x dx=\frac{7x^{2+1}}{2+1}-\frac{x^{3+1}}{3+1}+2(\frac{x^{1+1}}{1+1})$$
∴$$\int 7x^2-x^3+2x dx=\frac{7x^3}{3}-\frac{x^4}{4}+x^2+C$$

3. Find the integral of 2 sin⁡2x+3.
a) sin⁡2x+3x+C
b) -cos⁡2x-3x3+C
c) -cos⁡2x+3x+C
d) cos⁡2x-3x+12+C

Explanation: To find ∫ 2 sin⁡2x+3 dx
$$\int \,2 \,sin⁡2x+3 \,dx=\int \,2 \,sin⁡2x \,dx + \int \,3 \,dx$$
$$\int \,2 \,sin⁡2x+3 \,dx=2\int \,sin⁡2x \,dx+3\int \,dx$$
$$\int \,2 \,sin⁡2x+3 \,dx=\frac{-2 cos⁡2x}{2}+3x$$
∴∫2 sin⁡2x+3 dx=-cos⁡2x+3x+C
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4. Find the integral of $$\int 3e^x+\frac{2}{x}+x^3 dx$$.
a) $$3e^3x+\frac{2}{x}-\frac{x^4}{4}+c$$
b) $$3e^x+2 \,log⁡x+\frac{x^4}{4}+c$$
c) $$e^x+2 \,log⁡x+\frac{x^4}{4}+c$$
d) $$3e^x-\frac{2}{x^2}+\frac{x^4}{4}+c$$

Explanation: To find $$\int \,3e^x+\frac{2}{x}+x^3 \,dx$$
$$\int \,3e^x+\frac{2}{x}+x^3 dx=3\int \,e^x \,dx+2\int \frac{1}{x} \,dx+\int x^3 \,dx$$
$$\int \,e^x \,dx=e^x$$
$$\int \frac{1}{x} dx=log⁡x$$
∴$$\int 3e^x+\frac{2}{x}+x^3 \,dx=3e^x+2 \,log⁡x+\frac{x^4}{4}+c$$

5. Find the integral of $$\frac{4x^4-3x^2}{x^3}$$.
a) 7x2-3 log⁡x3+C
b) 2x2-3 log⁡x+C
c) x2-log⁡x+C
d) 2x2+3 log⁡x+C

Explanation: To find $$\int \frac{4x^4-3x^2}{x^3} dx$$
$$\int \frac{4x^4-3x^2}{x^3} \,dx=\int \frac{4x^4}{x^3} – \frac{3x^2}{x^3} \,dx$$
$$\int \frac{4x^4-3x^2}{x^3} \,dx=\int 4x dx-\int \frac{3}{x} dx$$
$$\int \frac{4x^4-3x^2}{x^3} \,dx=\frac{4x^2}{2}-3 log⁡x$$
∴ $$\int \frac{4x^4-3x^2}{x^3} \,dx=2x^2-3 \,log⁡x+C$$.

6. Find $$\int \,3 \,cos⁡x+\frac{1}{x} dx$$.
a) $$3 \,sin⁡x-\frac{1}{x}+C$$
b) $$2 \,sin⁡x+\frac{1}{x^3}+C$$
c) $$3 \,sin⁡3x+\frac{1}{x}+C$$
d) $$sin⁡x-\frac{1}{x^2}+C$$

Explanation: To find $$\int \,3 \,cos⁡x+\frac{1}{x^2} dx$$
$$\int \,3 \,cos⁡x+\frac{1}{x^2} dx=3 \int cos⁡x \,dx+\int \frac{1}{x^2} \,dx$$
$$\int \,3 \,cos⁡x+\frac{1}{x^2} dx=3 \,sin⁡x+\int x^{-2} \,dx$$
$$\int \,3 \,cos⁡x+\frac{1}{x^2} dx=3 \,sin⁡x+\frac{x^{-2+1}}{-2+1}$$
$$\int \,3 \,cos⁡x+\frac{1}{x^2} dx=3 \,sin⁡x-\frac{1}{x}+C$$

7. Find $$\int (2+x)x\sqrt{x} dx$$.
a) $$\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{9}+C$$
b) $$\frac{4x^{5/2}}{5}-\frac{2x^{7/2}}{7}+C$$
c) $$\frac{4x^{5/2}}{6}+\frac{2x^{7/2}}{7}+C$$
d) –$$\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{7}+C$$

Explanation: To find $$\int (2+x)x\sqrt{x} dx$$
$$\int \,(2+x)x\sqrt{x} \,dx=\int \,2x\sqrt{x}+x^{5/2} \,dx$$
$$\int \,(2+x)x\sqrt{x} \,dx=\int \,2x^{3/2} dx + \int x^{5/2} dx$$
$$\int \,(2+x)x\sqrt{x} \,dx=\frac{2x^{3/2+1}}{3/2+1}+\frac{x^{5/2+1}}{5/2+1}$$
$$\int \,(2+x)x\sqrt{x} \,dx=\frac{4x^{5/2}}{5}+\frac{2x^{7/2}}{7}+C$$

8. Find $$\int \,7x^8-4e^{2x}-\frac{2}{x^2} \,dx$$.
a) $$\frac{7x^4}{4}-2e^{2x}+\frac{2}{x}+C$$
b) $$\frac{7x^4}{4}+2e^{2x}+\frac{2}{x}+C$$
c) $$\frac{7x^4}{4}-2e^{2x} \frac{2}{x^2}+C$$
d) $$\frac{7x^4}{8}+2e^{2x}-\frac{4}{x}+C$$

Explanation: To find:$$\int 7x^8-4e^{2x}-\frac{2}{x^2} dx$$
$$\int \,7x^8-4e^{2x}-\frac{2}{x^2} \,dx=\int 7x^9 dx-4\int e^{2x} dx-2\int \frac{1}{x}^2 dx$$
$$\int \,7x^8-4e^{2x}-\frac{2}{x^2} \,dx=\frac{7x^{9+1}}{9+1}-\frac{4e^{2x}}{2}-\frac{2x^{-2+1}}{-2+1}$$
∴$$\int \,7x^8-4e^{2x}-\frac{2}{x^2} dx=\frac{7x^{10}}{10}-2e^{2x}+\frac{2}{x}+C$$

9. Find the integral $$\int sin⁡2x+e^3x-cos⁡3x dx$$.
a) –$$\frac{sin⁡2x}{2}+\frac{e^{3x}}{3}-\frac{sin⁡3x}{3}+C$$
b) –$$\frac{cos⁡2x}{2}+\frac{e^{3x}}{3}-\frac{sin⁡3x}{3}+C$$
c) $$\frac{cos⁡2x}{2}+\frac{e^{3x}}{3}-\frac{cos⁡3x}{3}+C$$
d) –$$\frac{cos⁡2x}{2}-\frac{e^{3x}}{3}+\frac{cos⁡3x}{3}+C$$

Explanation: To find $$\int \,sin⁡2x+e^{3x}-cos⁡3x \,dx$$
$$\int sin⁡2x+e^{3x}-cos⁡3x \,dx=\int \,sin⁡2x \,dx+\int \,e^{3x} \,dx-\int \,cos⁡3x \,dx$$
$$\int sin⁡2x+e^{3x}-cos⁡3x \,dx=-\frac{cos⁡2x}{2}+\frac{e^{3x}}{3}-\frac{sin⁡3x}{3}+C$$

10. Find the integral of (ax2+b)2.
a) $$\frac{a^2 \,x^5}{5}+b^2 \,x+\frac{2abx^3}{3}+C$$
b) –$$\frac{a^2 \,x^5}{5}-b^2 \,x+\frac{2abx^3}{3}+C$$
c) $$\frac{b^2 \,x^5}{5}+b^2 x+\frac{27x^3}{3}+C$$
d) $$\frac{a^2 \,x^5}{5}+x+\frac{2abx^3}{5}+C$$

Explanation: To find (ax2+b)2
$$\int (ax^2+b)^2 dx=\int (a^2 \,x^4+b^2+2ax^2 \,b) dx$$
$$\int (ax^2+b)^2 dx=\int \,a^2 \,x^4 \,dx+\int \,b^2 \,dx+2\int \,ax^2 \,b \,dx$$
$$\int (ax^2+b)^2 dx=a^2 \,\int \,x^4 \,dx+b^2 \int \,dx+2ab\int \,x^2 \,dx$$
$$\int (ax^2+b)^2 dx=a^2 (\frac{x^5}{5})+b^2 x+2ab(\frac{x^3}{3})$$
$$\int (ax^2+b)^2 dx=\frac{a^2 \,x^5}{5}+b^2 x+\frac{2abx^3}{3}+C$$

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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