Calculate the following integrals:

Exercise 1

\int cos x \sqrt{sin x} dx

 

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Exercise 2

\int sec^3x tan x dx

 

Exercise 3

\int \frac{dx}{\sqrt{x} cos^2 \sqrt{x}}

 

Exercise 4

\int \frac{dx}{x(1 + ln x)^3}

 

Exercise 5

\int \frac{sin^2x}{cos^4x}dx

 

Exercise 6

\int \frac{sin x + tan x}{cos x}dx

 

Exercise 7

\int \frac{dx}{sin x cos x}

 

Exercise 8

\int \sqrt{x \sqrt{x}} dx

 

Exercise 9

\int \sqrt[3]{x \sqrt{\frac{2}{x}}}dx

 

Exercise 10

\int \frac{cos x}{\sqrt{sin^3x}} dx

 

Exercise 11

\int \frac{sinx - cos x}{sin x+ cos x} dx

 

Exercise 12

\int \frac{dx}{(1 + x^2) arctanx}

 

Exercise 13

\int \frac{sec^2x}{1 + tan^2 x}dx

 

Exercise 14

\int \frac{(2 ln x)^2} {4x} dx

 

Exercise 15

\int \frac{lnx^2}{x} dx

 

Exercise 16

\int \frac{dx}{\sqrt{x} cos^2 \sqrt{x}}

 

Exercise 17

\int \frac{4^x + 5 \cdot 16^x}{1 + 16^x} dx

 

Exercise 18

\int \frac{tan \sqrt{x}} {\sqrt{x}} dx

 

Exercise 19

\int \frac{\sqrt{7 + 2 tan x}} {cos^2 x} dx

 

Exercise 20

\int \frac{dx}{x\sqrt{1 - ln^2x}}

 

 

Solution of exercise 1

\int cos x \sqrt{sin x}dx

=\int (sin x)^{\frac{1}{2}} cos x dx = \frac{(sin x) ^{\frac{3}{2}}} {\frac{3}{2}} + C

=\frac{2}{3} sin x \sqrt{sin x} + C

Solution of exercise 2

\int sec^3 x tan x dx

y = sec x

y' = sec x tan x

\int sec^2 x sec x tan x dx = \frac{1}{3} sec^3 x + C

Solution of exercise 3

\frac{dx}{\sqrt{x} cos^2 \sqrt{x}}

= 2 \int \frac{\frac{1}{2\sqrt{x}}} {\sqrt{x} cos^\sqrt{x}}} dx = 2 tan \sqrt{x} + C

Solution of exercise 4

\int \frac{dx}{x (1 + ln x)^3}

= \int (1 + ln x) ^{-3} \frac{1}{x} dx = \frac{(1 + ln x)^{-2}} {2} + C

= - \frac{1}{2 (1 + lnx)^2} + C

Solution of exercise 5

\int \frac{sin^2x}{cos^4x} dx

= \int \frac{sin^2 x}{cos^2x} \frac{1}{cos^2x}dx =\int tan^2x \frac{1}{cos^2x} dx

= \frac{1}{3} tan^3x + C

Solution of exercise 6

\int \frac{sinx + tan x}{cos x} dx

= \int \frac{sin x}{cos x} dx + \int \frac{tanx}{cos x} dx

= \int \frac{sin x}{cos x}dx + \int tanx sec x dx = - ln (cos x) + sec x + C

Solution of exercise 7

\int \frac{dx}{sin x cos x}

\int \frac{sin^2x + cos^2x}{sin x cos x} dx = \int \frac{sin x}{cos x} dx + \int \frac{cos x}{sin x} dx

= - ln (cos x) + ln (sin x) + C = ln (tan x) + C

Solution of exercise 8

\int \sqrt{x \sqrt{x}} dx

\int \sqrt{x \sqrt{x}} dx \int \sqrt{\sqrt{x^2 \cdot x}} dx = \int \sqrt[4]{x^3}dx = \int x ^{\frac{3}{4}} dx

= \int \frac{x ^{\frac{3}{4} + 1}} {\frac{3}{4} + 1} dx

= \frac{4}{7} x^{\frac{7}{4}} + C = \frac{4}{7} \sqrt[4]{x^7} + C

= \frac{4}{7} x \sqrt[4]{x^3} + C

Solution of exercise 9

\int \sqrt[3]{x \sqrt{\frac{2}{x}}}dx

= \int \sqrt[3] {\sqrt{\frac{2x^2}{x}}} dx

= \int \sqrt[6]{2x} dx

= \sqrt[6]{2} \int x^{\frac{1}{6}} dx

=\sqrt[6]{2} \int \frac{x^{\frac{1}{6} + 1}} {\frac{1}{6} + 1} dx

= \frac{6}{7} \sqrt[6]{2} x^{\frac{7}{6}} + C

= \frac{6}{7} \sqrt[6]{2} \sqrt[6]{x^7} + C

= \frac{6}{7} x \sqrt[6]{2x} + C

Solution of exercise 10

\int \frac{cos x}{\sqrt{sin^3x}} dx

= \int cosx sen ^{-\frac{3}{2}}x dx

= \frac{sin^{-\frac{1}{2}}x}{-\frac{1}{2}} + C

= \frac{-2}{\sqrt{sin x}} + C

Solution of exercise 11

\int \frac{sinx - cos x}{sin x+ cos x} dx

= - \int \frac{cos x - sin x}{sin x+ cos x} dx

= - ln (sin x + cos x) + C

Solution of exercise 12

\int \frac{dx}{(1 + x^2) arctanx}

= \frac{\frac{1}{1 + x^2}} {arctan x} dx

= ln (arc tag x) + C

Solution of exercise 13

\int \frac{sec^2x}{1 + tan^2 x}dx

= arc tan(tan x) + C = x + C

Solution of exercise 14

\int \frac{(2 ln x)^2} {4x} dx

= \int ln^2 (x) \frac{1}{x}dx = \frac{1}{3} ln^3 x + C

Solution of exercise 15

\int \frac{lnx^2}{x} dx

= \int 2 ln (x) \frac{1}{x}dx

=ln^2 x + C

Solution of exercise 16

\int \frac{dx}{\sqrt{x} cos^2 \sqrt{x}}

= 2 \int \frac{dx}{cos^2 \sqrt{x}} \frac{1}{2 \sqrt{x}} dx = 2 tan \sqrt{x} + C

Solution of exercise 17

\int \frac{4^x + 5 \cdot 16^x}{1 + 16^x} dx

= \int \frac{4^x}{1 + (4^x)^2} dx + 5 \int \frac{4 ^{2x}} {1 + 4^{2x}} dx

= \frac{1}{ln4} arc tg (4^x) + \frac{5}{2 ln 4} ln (1 + 4 ^{2x}) + C

Solution of exercise 18

\int \frac{tan \sqrt{x}} {\sqrt{x}} dx

= -2 \int \frac{-sin \sqrt{x}} {cos \sqrt{x}} \frac{1}{2\sqrt{x}}dx = -2 ln (cos \sqrt{x}) + C

Solution of exercise 19

\int \frac{\sqrt{7 + 2 tan x}} {cos^2 x} dx

= \frac{1}{2} \int (7 + 2 tan x)^{\frac{1}{2}} \frac{2}{cos^2x} dx

= \frac{1}{2} \frac{(7 + 2 tanx) ^ {\frac{3}{2}}} {\frac{3}{2}} + C = \frac{1}{3} (7 + 2 tan x) \sqrt{(7 + 2 tan x)} + C

Solution of exercise 20

\int \frac{dx}{x\sqrt{1 - ln^2x}}

= \int \frac{\frac{1}{x}} {\sqrt{1 - ln^2x}} dx = arc sin(ln x) + C

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.