What is the integral from 0 to 1.5 of 1/(1+sin(x)) ?

The integral from 0 to 1.5 of 1/(1+sin(x)) is 0.96458…

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 0 to 1.5 of 1/(1+sin(x))

\int_{0}^{1.5} \frac{1}{1 + sin ( x )} d x

0.96458 \dots

Solution steps

\int_{0}^{1.5} \frac{1}{1 + sin ( x )} d x

Apply u-substitution

= \int_{0}^{0.93159 \dots} \frac{2}{1 + u ^{2} + 2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{0.93159 \dots} \frac{1}{1 + u ^{2} + 2 u} d u

Complete the square

1 + u^{2} + 2 u : (u + 1)^{2}

= 2 \cdot \int_{0}^{0.93159 \dots} \frac{1}{( u + 1 ) ^{2}} d u

Apply u-substitution

= 2 \cdot \int_{1}^{1.93159 \dots} \frac{1}{v ^{2}} d v

Apply the Power Rule

= 2 [- \frac{1}{v}]_{1}^{1.93159 \dots}

Compute the boundaries:

0.48229 \dots

= 2 \cdot 0.48229 \dots

Simplify

= 0.96458 \dots

\int_{0}^{1} x^{3} 7 + 5 x^{4} d x

\int_{0}^{1} (x - 6) e^{x} d x

\int_{- 1}^{2} (x + 4) - (x^{2} + 2) d x

\int_{- 8}^{5} (1 - x) d x

\int_{2}^{10} 1

What is the integral from 0 to 1.5 of 1/(1+sin(x)) ?
The integral from 0 to 1.5 of 1/(1+sin(x)) is 0.96458…