Question Statement

Evaluate the integral: $\int (cos x)^{\frac{1}{5}} sin x d x$

Background and Explanation

This problem involves recognizing the pattern for integrating a composite function where the integrand contains a function raised to a power multiplied by its derivative (or a scalar multiple thereof). The key insight is identifying that $sin x$ is related to the derivative of $cos x$ .

Solution

Let $I$ denote the integral: $I = \int (cos x)^{1/5} \cdot sin x d x$

To apply the standard power rule for integration, we need the derivative of the base function $cos x$ , which is $- sin x$ . Since we have $+ sin x$ in the integrand, we factor out $- 1$ to create the proper form:

$I = (- 1) \int (cos x)^{1/5} \cdot (- sin x) d x$

Now we apply the power rule for integration: $\int [f (x)]^{n} \cdot f^{'} (x) d x = \frac{[ f ( x ) ] ^{n + 1}}{n + 1} + C$

Here, $f (x) = cos x$ , $f^{'} (x) = - sin x$ , and $n = \frac{1}{5}$ . Substituting these values:

$I = (- 1) \cdot \frac{( c o s x ) ^{\frac{1}{5} + 1}}{\frac{1}{5} + 1} + c$

Simplifying the exponents and denominator: $I = (- 1) \cdot \frac{( c o s x ) ^{\frac{6}{5}}}{\frac{6}{5}} + c$

Dividing by the fraction $\frac{6}{5}$ (which is equivalent to multiplying by $\frac{5}{6}$ ): $I = - \frac{5}{6} (cos x)^{\frac{6}{5}} + c$

This result can also be written using alternative notation for the exponent: $I = - \frac{5}{6} cos^{\frac{6}{5}} x + c$

Key Formulas or Methods Used

Power Rule for Composite Functions: $\int [f (x)]^{n} \cdot f^{'} (x) d x = \frac{[ f ( x ) ] ^{n + 1}}{n + 1} + C$ (valid for $n \neq = - 1$ )
Derivative of Cosine: $\frac{d}{d x} (cos x) = - sin x$
Sign Adjustment: Factoring out $- 1$ to match the derivative pattern: $sin x = - (- sin x)$

Summary of Steps

Identify the pattern: Recognize that $sin x$ is related to the derivative of $cos x$ , specifically $\frac{d}{d x} (cos x) = - sin x$
Adjust the sign: Rewrite the integral as $I = - \int (cos x)^{1/5} \cdot (- sin x) d x$ to match the form $\int [f (x)]^{n} f^{'} (x) d x$
Apply the power rule: Increase the exponent by 1 and divide by the new exponent: $- \frac{( cos x ) ^{6/5}}{6/5} + C$
Simplify: Multiply by the reciprocal of the denominator to obtain $- \frac{5}{6} (cos x)^{6/5} + C$

Solution

Let

I

denote the integral:

I = \int (cos x)^{1/5} \cdot sin x d x

To apply the standard power rule for integration, we need the derivative of the base function

cos x

, which is

- sin x

. Since we have

+ sin x

in the integrand, we factor out

- 1

to create the proper form:

I = (- 1) \int (cos x)^{1/5} \cdot (- sin x) d x

Now we apply the power rule for integration:

\int [f (x)]^{n} \cdot f^{'} (x) d x = \frac{[ f ( x ) ] ^{n + 1}}{n + 1} + C

Here,

f (x) = cos x

f^{'} (x) = - sin x

, and

n = \frac{1}{5}

. Substituting these values:

I = (- 1) \cdot \frac{( c o s x ) ^{\frac{1}{5} + 1}}{\frac{1}{5} + 1} + c

Simplifying the exponents and denominator:

I = (- 1) \cdot \frac{( c o s x ) ^{\frac{6}{5}}}{\frac{6}{5}} + c

Dividing by the fraction

\frac{6}{5}

(which is equivalent to multiplying by

\frac{5}{6}

I = - \frac{5}{6} (cos x)^{\frac{6}{5}} + c

This result can also be written using alternative notation for the exponent:

I = - \frac{5}{6} cos^{\frac{6}{5}} x + c

Summary of Steps

Identify the pattern: Recognize that

sin x

is related to the derivative of

cos x

, specifically

\frac{d}{d x} (cos x) = - sin x

Adjust the sign: Rewrite the integral as

I = - \int (cos x)^{1/5} \cdot (- sin x) d x

to match the form

\int [f (x)]^{n} f^{'} (x) d x

Apply the power rule: Increase the exponent by 1 and divide by the new exponent:

- \frac{( cos x ) ^{6/5}}{6/5} + C

Simplify: Multiply by the reciprocal of the denominator to obtain

- \frac{5}{6} (cos x)^{6/5} + C

3.2 Q-16

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

3.2 Q-16

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps