Evaluate the definite integral: integral_0^{pi / 2} cos x / {1 + sin x} dx.

Question:

Evaluate the definite integral:

{eq}\displaystyle \int\limits_0^{\dfrac \pi 2} \dfrac {\cos x} {1 + \sin x}\ dx {/eq}.

Definite Integration

The fundamental theorem of calculus states that {eq}\displaystyle \int_a^b f(x)dx = F(b)-F(a) {/eq} where {eq}\displaystyle \int f(x)dx = F(x) {/eq}

In the method of integration by substitution, the variable is substituted with another function so that the complicated integral becomes easier.

Some standard integrals are {eq}\displaystyle \int \frac{f'(x)}{f(x)}dx = \ln |f(x)|+C , \int \frac{1}{t} dt = \ln t +C {/eq} where {eq}\displaystyle C {/eq} is the constant of integration.

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Let {eq}\displaystyle I =\int_0^{\frac{\pi}{2}} \frac{ \cos x}{1+ \sin x } dx {/eq}

Put {eq}\displaystyle \sin x =t {/eq}

Differentiating, we... Evaluating Definite Integrals Using the Fundamental Theorem

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Chapter 16 / Lesson 2
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In calculus, the fundamental theorem is an essential tool that helps explain the relationship between integration and differentiation. Learn about evaluating definite integrals using the fundamental theorem, and work examples to gain understanding.