Evaluate the integrals in Exercises 87–96.
93. ∫₀^(π/2) 7^(cos t) sin t dt

검증된 단계별 안내

Recognize that the integral is of the form $\int_0^{\frac{\pi}{2}} 7^{\cos t} \sin t \, dt$, which suggests a substitution involving the exponent's inner function $\cos t$.

Let $u = \cos t$. Then, compute the differential $du = -\sin t \, dt$, which implies $\sin t \, dt = -du$.

Change the limits of integration according to the substitution: when $t = 0$, $u = \cos 0 = 1$; when $t = \frac{\pi}{2}$, $u = \cos \frac{\pi}{2} = 0$.

Rewrite the integral in terms of $u$: $\int_0^{\frac{\pi}{2}} 7^{\cos t} \sin t \, dt = \int_{u=1}^{u=0} 7^u (-du) = \int_0^1 7^u \, du$.

Now, integrate \$7^u$ with respect to $u$ using the formula $\int a^x \, dx = \frac{a^x}{\ln a} + C$, and then apply the new limits from 0 to 1.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integrals

A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves setting a new variable equal to a function inside the integral, then rewriting the integral in terms of this variable and its differential. This technique is especially useful when the integral contains a composite function.

Exponential Functions with Variable Exponents

Exponential functions with variable exponents, such as 7^(cos t), require careful handling during integration. Recognizing that 7^(cos t) can be rewritten using the natural exponential function as e^(cos t * ln 7) helps in applying substitution or other integration techniques effectively.

Evaluate the integrals in Exercises 87–96.93. ∫₀^(π/2) 7^(cos t) sin t dt